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Find All Maximal Cliques , get all 3cliques, CLQ 3 (by applying all CS0 deletions)

3 3 6 4 8 8 a e c 5 4 6 3 3 Count. AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18. Research (my view):

garysmith
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Find All Maximal Cliques , get all 3cliques, CLQ 3 (by applying all CS0 deletions)

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  1. 3 3 6 4 8 8 a e c 5 4 6 3 3 Count AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Research (my view): • Have a research program (what big contribution you want to make to advance the body of current knowledge). Be bold about that! There is nothing you can’t do! • Know the big and the little pictures (lots of reading and studying of what’s been done and how). Start with the Plenary Talks of the big shots at 1st rate recent conferences or survey publications. You don’t have to agree with them but you do have to be as knowledgeable as they are. A big disagreement IS a program! • Go back and forth between big picture and details; telescope and microscope; forest and trees. • Your program is your intended contribution to the big picture/forest. Individual papers (not surveys) are your detailed contributions on the path to your program goal. • Be willing to present at major conferences and be willing to get beat up! That’s how we hone our skills and our program. And, humiliation is a great motivator. • Think big. That takes money. Go after big grants; endowed chairs, etc. Develop pipelines to very good foreign universities. Contribute to Conference management, etc. • Always look for people to work for you who are smarter than you are. Never be afraid to hire such people. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 • A 3cliques that shares an edge with a 3clique form a 4clique iff the 6th unknown edge exists. Two 3cliques that share a vertex never form a 4clique (The can form a 5clique, but that 5clique will be discovered in 3. below). 3. A 4clique and a 3clique that shares an edge form a 5clique iff the 9th and 10th unknown edges exist. My research program is to manage and mine big, complex data. Big means “lots of objects (rows (and sometimes columns) in tables, edges in graphs). Complex data seems to mean graphs. So I’m interested in tables with lots of rows (including edge tables of graphs). My main tool is horizontal processing. That’s my forest. ANalystTickerSymbolRelationship with labels (1=“recommends”) H B B SB B SS S S H H B B B SB Buy-Hold-Sell 1 3 1 4 4 1 2 3 1 1 2 2 3 4 SA 4. A 5clique and a 3clique that shares an edge form a 6clique iff the 13th 14th and 15th unknown edges exist. A universal data structure for my work is the vertex-labelled, edge-labelled graph, e.g., SA 8 5 8 8 2 1 8 8 5 7 4 5 6 3 9 9 6 7 dow 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ct 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 sal 9 5 8 8 2 1 8 8 5 7 4 5 6 3 9 9 6 7 F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 We can think about this structure many ways, as a relationship with entity tables; as a AN[lysist] Table in which the TickerSymbol pTrees are additional attributes; as a TS Table in which the AN pTrees are additional attributes. 1 2 3 1 2 3 4 5 4 3 5 4 3 2 5 1 3 1 2 4 3 1 2 As a relationship with entity tables attached we can expand to other relationships sharing one or more of the entities (the RoloDex Model). More comprehensive, the graph could be 3D, 4D (i.e., edges are triples, quadruples), etc. Find All Maximal Cliques, get all 3cliques, CLQ3 (by applying all CS0 deletions) 1. If a 3cliques shares nothing with any other 3clique, then it is maximal, else: We find all 3cliques with CS0). Two 3cliques can share: a, nothing b. One vertex and no edges c. Two verticies (them 1 edge also) • A Kcliqueand 3clique that shares an edge form a (K+1)clique iff all K-2 edges connecting the non-shared Kclique and 3clique vertex exist (to add to the K(K-1)/2 edges of the Kclique) 4 Md pointed out that b is covered by c since for b to form a 5clique, the edge (4,1) would have to exist, in which case 1345 and 1234 both are examples of two 3cliques sharing and edge (2, above), so we need only do 1,2,3,4,5. As soon as we reach a K such that there is no 3clique sharing and edge with the kclique then we know that that Kclique is maximal. At that point we remove from the list of 3cliques all 3cliques that were included in the maximal Kclique just found, and start over.

  2. A MaxClique Thm: A Kclique and a 3clique that shares an edge form a (K+1)clique iff All K-2 edges from the non-shared Kclique vertices to the non-shared 3clique vertex exist in the graph. G7 Clique Induction Algorithm: Let CLQK be the set of all Kcliques, 1st find CLQ3 using CS0. Induction Step: CLQK+1 is obtained by applying MCT to CLQK and CLQ3. Applying Clique Induction Algorithm to G7: 1 2 3 18,20,22CLQ4 since 3:18,20,22E3 1 1 1 3 3 3 2 4 2 4 Note checkback. Is it required? (No: if 132 in 4CLQ it’d show up already). Already in CLQ4 E 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 2 3 4 3 4 since 34E3 1 3 4 8 1 2 3 8 1 2 3 14 1 2 4 8 1 2 4 14 UCLQ4 done. Is a pTree version faster? 1 3 4 14 2 3 4 8 2 3 4 14 18 22 20 14 2 12 4 3 1 8 16 15 17 31 27 10 23 19 21 34 30 33 11 24 29 28 32 25 26 7 6 5 9 13 All already contained UCLQ5 1 3 9 1 4 13 1 6 7 1 6 11 1 2 20 1 2 22 1 5 7 1 5 11 MUCLQs 1 2 18 3 9 33 1 2 3 4 8 1 2 3 4 14 1 2 3 4 8 1 2 3 4 14 9 31 33 6 7 17 UCLQ3 Unique 3cliques as lists 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 8 14 18 20 22 4 8 9 14 1 1 1 1 1 1 1 2 2 2 4 4 4 5 5 6 6 3 3 3 8 13 14 7 11 7 11 4 8 14 2 2 3 3 3 6 9 9 4 4 4 4 9 7 31 31 8 14 8 14 33 17 33 34 24 24 24 25 27 29 28 30 30 26 30 32 34 33 34 32 34 34 9 31 34 24 28 34 24 30 33 24 30 34 25 26 32 27 30 34 29 32 34 MUCLQ Clustering Alg? Combine iff share all but 1 1 2 3 4 8 14 1 3 9 33 1 4 13 1 6 7 11 1 2 18 20 22 1 5 7 11 6 7 17 9 31 33 34 24 28 34 24 30 33 34 25 26 32 27 30 34 29 32 34 Combine iff share all but 1, round 2 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 1 2 3 4 8 14 1 2 18 20 22 1 3 9 33 1 5 7 11 1 4 13 9 31 33 34 24 28 34 24 30 33 34 25 26 32 27 30 34 29 32 34 6 7 17 CLQ3 (as pTrees) Remaining edges after CS0 (removal of PURE0 edge endpoint pair ANDs). 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 111314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 28303334 2632 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 6 Combine iff share all but 1 rn3 no change Rnd4: Combine 3CLQs if match all but 1 1 2 3 4 8 14 1 2 18 20 22 1 3 9 33 9 31 33 34 24 28 34 24 30 33 34 25 26 32 27 30 34 29 32 34 1 5 7 11 1 5 7 11 6 6 17 13 Rnd5: Combine 3,4CLQs if match all but 2 1 2 3 4 8 14 1 2 18 20 22 1 3 9 33 24 28 34 25 26 32 27 30 34 29 32 34 30 33 31 34 17 13 Doesn’t seem very good!

  3. Find a Maximal Maximal-Clique for each v (a MaxClique containing v with max # of vertices). In the Kclique Induction Theorem we checked edges connecting non-shared vertices. Here we use a Kclique APRIORI Algorithm which may be faster since, e.g., a candidate 4clique which survives the “all sub3sets are 3ciques” is automatically a 4clique. G7 Cand4Cliques 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 8 14 18 20 22 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 8 8 8 14 14 18 18 20 8 14 18 20 22 14 20 22 20 22 20 22 22 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4 4 5 6 4 4 4 8 8 9 8 8 13 7 7 8 9 14 9 14 14 13 14 14 11 11 2 2 2 2 3 9 24 3 3 3 4 4 31 30 4 4 8 8 8 33 33 8 14 14 14 14 34 34 No LAST 3 no 3 8 9,14 no 6 7 11 no 4 13 14 no 4 8 14 no 5 7 11 no 3 4 9 no 3 9 14 no 4 8 13 no 2 3 20 no 2 4 20 no 2 4 22 no 2 4 18 no 2 8 14,20,22 no 2 3 18 no 2 3 22 No LAST 3 E 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Survivor 4Cliqs 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 8 4 4 4 8 14 8 14 8 14 8 14 Surviv5Cliqs 1 1 2 2 3 3 4 4 8 14 Can6Clqs 1 2 3 4 8 14 16 9 10 6 3 4 4 4 5 2 3 1 2 52 2 2 2 2 32 22 53 3 2 43 4 4 6 11 16 We have 3 clique mining methods,: 1 This list APRIORI method. 2 The pTree version o f this APRIORI 22 18 20 14 4 1 2 3 12 8 21 31 10 33 19 23 30 16 34 27 17 15 29 32 28 25 24 11 26 7 6 5 9 13 Unique 3Cliques (in set form) 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 8 14 18 20 22 4 8 9 14 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 5 5 6 6 3 3 3 4 4 8 13 14 7 11 7 11 4 8 14 8 14 3 3 3 6 9 9 4 4 9 7 31 31 8 14 33 17 33 34 24 24 24 25 27 29 28 30 30 26 30 32 34 33 34 32 34 34 3 We have this APRIORI Clique Mining Alg (list or pTree version) and the previous Induction Clique Mining Alg (list or pTree version). Which is fastest? Simplest. (Accuracy should be the same (100%). Do we need 100% or can we get great time savings by relaxing that? How do these methods perform on a Big Graph? On Friends? sh 1 2 3 1 2 4 8 14 2 3 4 8 14 Cand5Cliqs 1 1 1 2 2 2 3 3 3 4 4 8 8 14 14 no 2 3 4 8 14 no 3 4 8 14 no 2 4 8 14 no 2 3 8 14 On G7, 2 max5cliques 12348 and 1234e. If we only need 1 MaxSized MaxCliq for each vertex, we can now eliminate 12348e. All 4cliqs are subset of this 5cliq, so the MSMC(v) for all other vertices, v, one can chose at random from v’s 3Cliques. I choose 1st unique one (if one) MSMC(1,2,3,4,8,14)= {1,2,3,4,8,14} s1 2 4 2 3 4 Remaining pairwise ANDs after removal of PURE0s (i.e., after CS0). So these are the 3cliques in pTree form. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 111314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 28303334 2632 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 MSMC(5,7)= {1,5,7} MSMC(9,31,33)= {9,31,33} MSMC(10)= {3,10} MSMC(11)= {1,5,11} MSMC(12)= {1,12} MSMC(13)= {1,4,13} MSMC(*)= {33,*} *=15,16,19,21,23 MSMC(17)= {6,7,17} MSMC(18)= {1,4,18} MSMC(20)= {1,2,20} MSMC(22)= {1,2,22} MSMC(24,28,34)= {24,28,34} MSMC(25,26,32)= {25,26,32} MSMC(27,30)= {27,30,34} MSMC(27,30)= {27,30,34} MSMC(28)= {3,28} MSMC(29)= {29,32,34}

  4. G1 APPENDIX Divisive Graph Clustering: Girvan and Neuman delete edges with max “betweenness”, i.e., max participation in shortest paths (of all lengths). 2 2 4 3 3 2 2 3 2 1 1 1 1 G1_1 1 G1_2 G1_2 G1_3 4 4 5 5 5 CS0 Algorithm: Delete edge with zero Common Sibling (CS0) co-participation. The pTree calculation of CS(h,k)=E(h)&E(k) is instantaneous. We use CS0 on S1P=E only. S 1 P 1 0 1 0 0 0 1 S 1 P 1 0 1 0 0 0 1 S 1 P 5 0 1 0 0 0 1 S 1 P 5 0 1 0 1 0 2 S 1 P 2 1 0 1 1 1 4 S 1 P 2 1 0 1 0 1 3 S 1 P 3 0 1 0 0 0 1 S 1 P 3 0 1 0 1 0 2 S 1 P 4 0 0 1 0 1 2 S 1 P 4 0 1 0 0 0 1 S 1 P 1 0 0 1 0 1 S 1 P 1 0 0 1 1 2 S 1 P 2 0 0 0 1 1 S 1 P 2 0 0 0 1 1 S 1 P 3 1 0 0 1 2 S 1 P 3 1 0 0 1 2 S 1 P 4 0 1 1 0 2 S 1 P 4 1 1 1 0 3 3 3 4 4 S 1 P 1 & 3 0 0 0 1 1 S 1 P 2 & 4 0 0 0 0 0 S 1 P 1 & 4 0 0 1 0 1 S 1 P 3 & 4 1 0 0 0 1 S 1 P 1 & 2 0 0 0 0 0 0 S 1 P 2 & 5 0 0 0 0 0 0 S 1 P 2 & 3 0 0 0 0 0 0 S 1 P 3 & 3 0 0 0 0 0 0 S 1 P 4 & 5 0 0 0 0 0 0 S 1 P 1 & 2 0 0 0 0 0 0 S 1 P 1 & 3 0 0 0 0 0 S 1 P 2 & 4 0 0 0 0 0 S 1 P 2 & 4 0 0 0 0 0 0 S 1 P 3 & 4 1 0 0 0 0 S 1 P 2 & 3 0 0 0 0 0 0 S 1 P 2 & 5 0 0 0 0 0 0 CS0 picks 24. Correct. CS0 says all edges are equal (correct?). CS0 says all edges are equal (seems correct). CS0 says all edges are equal (seems correct). CS0 picks 23 46 correctly CS0 picks 23 correctly. CS0 says all edges are equal. 6 6 6 6 F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 6 1 0 0 0 0 0 1 F A 1 1 2 0 0 0 0 0 1 1 F A 1 1 6 0 1 0 0 0 0 1 F A 1 3 4 0 0 0 0 1 0 1 F A 1 3 5 0 0 0 1 0 0 1 F A 1 4 5 0 0 1 0 0 0 1 F A 1 4 6 0 0 0 0 0 0 0 F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 6 1 0 0 0 0 0 1 F A 1 1 2 0 0 0 0 0 1 1 F A 1 1 6 0 1 0 0 0 0 1 F A 1 3 4 0 0 0 0 1 0 1 F A 1 3 5 0 0 0 1 0 0 1 F A 1 4 5 0 0 1 0 0 0 1 S 1 P 1 0 1 0 0 0 1 2 S 1 P 1 0 1 0 0 0 1 2 S 1 P 1 0 1 0 0 0 1 2 S 1 P 1 0 1 0 0 1 1 2 S 1 P 5 0 1 0 1 0 0 2 S 1 P 5 1 0 1 1 0 0 2 S 1 P 5 0 0 1 1 0 0 2 S 1 P 5 0 0 1 1 0 0 2 S 1 P 6 1 1 0 1 0 0 2 S 1 P 6 1 0 1 0 0 0 2 S 1 P 6 1 1 0 0 0 0 2 S 1 P 6 1 1 0 1 0 0 2 S 1 P 2 1 0 1 0 0 1 3 S 1 P 2 1 0 1 0 1 0 3 S 1 P 2 1 0 1 0 0 1 3 S 1 P 2 1 0 1 0 0 1 3 S 1 P 3 0 1 0 1 1 0 3 S 1 P 3 0 1 0 1 1 0 3 S 1 P 3 0 1 0 1 1 0 3 S 1 P 3 0 1 0 1 0 1 3 S 1 P 4 0 0 1 0 1 1 2 S 1 P 4 0 0 1 0 1 1 2 S 1 P 4 0 0 1 0 1 0 2 S 1 P 4 0 0 1 0 1 0 2 F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 5 0 0 0 0 0 0 0 F A 1 1 2 0 0 0 0 0 0 0 F A 1 1 6 0 0 0 0 0 0 0 F A 1 3 4 0 0 0 0 0 0 0 F A 1 3 6 0 0 0 0 0 0 0 F A 1 4 5 0 0 0 0 0 0 0 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 G1_4 G1_5 G1_7 G1_6 5 5 5 5 CS0 picks 15 23 46 correctly. F A 1 1 5 0 0 0 0 0 0 0 F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 6 1 0 0 0 0 0 1 F A 1 1 2 0 0 0 0 0 1 1 F A 1 1 6 0 1 0 0 0 0 1 F A 1 3 4 0 0 0 0 1 0 1 F A 1 3 5 0 0 0 1 0 0 1 F A 1 4 5 0 0 1 0 0 0 1 F A 1 4 6 0 0 0 0 0 0 0

  5. Divisive Graph Clustering CS0 on G7: Delete edge with the zero Common Siblings. S1P 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 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0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S1P pairwise ANDs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 14 1515 1616 1919 20 2121 2323 2424242424 252525 26 2727 28 2929 3030 313132 2 3 4 5 6 7 8 9 1112131418202232 3 4 8 1418202231 4 8 9 1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 333434 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 G7 CS0 says delete the zero counts above. 15 16 19 21 and 23 with 33 and 34 get deleted because they only have 33 and 34 as nbrs and 33 and 34 are not nbrs (i.e., they are friends with two enemies. They should not be deleted! Solution? (1,12) because 12 is only connected to 1. (1,32) correct. (2,31) correct. (20,34) correct. (3,10) correct. (24,26) and (25,28) are incorrect. But, recall that 24 and 28 are ambiguous wrt cluster? (3,28) correct. We can solve “delete if only connected to 1 pt” problem by checking the nbr count. (3,29) correct. (10,34) because, now, 10 is only connected to 34. The first round goes a long way toward splitting white-blue from green-yellow. (14,34) correct.

  6. Divisive Graph Clustering: What can be combined with CS0? Del CC0: (1,5) (1,6) (1,11) CS0-CC0: Unless it results in an isolated singleton or doubleton (keep 1,12) Delete all common Siblings=0 (CS0) and all common Cousins=0 (CC0). CC0: Delete edge(s) with zero Common 1st Cousins (CCh,kS2Ph & S2Pk). c d e Del 1,32 2,31 3,10 3,28 3,29 14,34 20,34 15,33 16,33 19,33 21,33 23,33 24,26 25,28 S2Ph= blue and orange This is CS0-CC0 b a So do the 1time SiblingANDs (S1Ph&S1Pk) and CousinANDs (S2Ph&S2Pk). Then in one pass reading counts CS0-CC0 deletes 12 edges (whereas Girvan-Neuman makes 1 pass per edge deletion and recalculates each new pass). Next we could delete more edges with our current counts or recaculate counts and redo CS0-CC0. h Use DelThresh=1 on Siblings (recalculating nothing): Delete additionally: 1,9 1,13 1,18 1,20 1,22 3,33 6,11 6,17 9,34 24,28 24,33 25,26 27,30 29,32 30,33 31,33 31,34 (but not 2,18 2,20 2,22 4,13 5,7 5,11 7,17 25,32 26,32 27,34 28,34 29,34; DONOT ISOLATE rule). This is CS1-CC0. k Use DelThresh=1 on Cousins: del 1,4 (but not 7,17 15,34 16,34 19,34 23,34 27,34 due to the DONOT ISOLATE rule.) . This is CS1-CC1. f g Likely, next round (after recalculating CS and CC), 1,7 and 3,9 will delete. Note: {10 15 16 19 21 23 24 27 28 29 30 34} has already separated as a component. Then the other clusters would be: {9 25 26 31 32 33} TheGreens TheYellows j i h S2Pk = red and green S2P pairwise ANDs 3 6 1 0 0 5 3 4 0 0 5 4 4 4 4 6 8 1212128 118 4 11111710189 102 151015121312121 1 188 2 4 1 5 1 5 1 5 1 2 5 6 5 1 1 13135 2 4 3 2 2 131 5 176 5 2 8 3 7 3 counts 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 101415151616191920212123232424242424252525262727282929303031313232 S2P-AND-OP-1 2 3 4 5 6 7 8 9 11121314182022323 4 8 14182022314 8 9 10142829338 13147 117 111717313334343433343334333434333433342628303334262832323034343234333433343334 S2P-AND-OP-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 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2424242424 252525 26 2727 28 2929 3030 313132 2 3 4 5 6 7 8 9 1112131418202232 3 4 8 1418202231 4 8 9 1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 333434 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 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0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

  7. (1,12) saved by the DNI rule. Unless singleton/doubleton is isolated, del CommonCousins0 and CommonSiblings2 Del CC0 (1,5) (1,6) (1,11) CS2-CC0 Del CS2 1:5,6,7,9,11,12,13,18,20,22,32 2:18,20,22,31 3:9,10,28,29,33 4:13 5:7,11 6:11,17 7:17 9:34 10:34 14:34 15:33,34 16:33,34 19:33,34 20:34 21:33,34 23:33,34 24:26,28,33 25:26,28,32 26:32 27:30,34 28:34 29:32.34 30:33 31:33.34 32:34 We get YellowGreen(-20) {20, 24, 28, 29 ,10,15,16,19,21,23,27,30,34)} {9, 31, 33,25,26,32} So again Black and Blue are a confused, but Yellow and Green are almost perfect. At this point we have looked at serveral threshold combinations for siblings and cousins. I think CS0-CC0 followed by a recalculation and then a reapplication of CS0-CC0 might be best. S2P pairwise ANDs 3 6 1 0 0 5 3 4 0 0 5 4 4 4 4 6 8 1212128 118 4 11111710189 102 151015121312121 1 188 2 4 1 5 1 5 1 5 1 2 5 6 5 1 1 13135 2 4 3 2 2 131 5 176 5 2 8 3 7 3 counts 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 101415151616191920212123232424242424252525262727282929303031313232 S2P-AND-OP-1 2 3 4 5 6 7 8 9 11121314182022323 4 8 14182022314 8 9 10142829338 13147 117 111717313334343433343334333434333433342628303334262832323034343234333433343334 S2P-AND-OP-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 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1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 333434 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

  8. CS0 and DONOT ISOLATE Note CS=0 deletion (CS0) will insure that we never break up a clique! Why? Every k-clique is made up of COMB)(k,3) and we never break up 3cliques – because we never delete gf. S1P(g) contains k,f and S1P(f) contains g,k so CS(g,f)=S1P(g)&S1P(f) contains k and therefore CS(g,f)  0. To insure we never break up cliques, for Round 1 we use “CS0 with DONOT ISOLATE rule” since it’s quick and has this nice clique preservation guarantee. 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 169 106 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 1116 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 17 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 22 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 28 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 29 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 31 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 32 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 33 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 34 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 k f g G7 Common Siblings and 3cliques Theorem: An edge, (h,k) has No Common Siblings (i.e., CSh,k ShSk= iff Eh&Ek is pure0) iff that edge is not involved in any 3clique. The proof is very simple: An edge (g,f) has common sibling, k, iff (g,f,k) is a 3clique. Thus, removing all edges with ZeroCommonSiblings leaves only 3cliques (of course, if the DONOT ISOLATE rule is in place, it leaves also leaves isolates.) Thus, instead of turning to CommonCousins (as we do on the next slide) maybe we ought to select pertinent 3cliques to break as a next step (which we do 2 slides ahead)? S1P pairwise ANDs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 14 1515 1616 1919 20 2121 2323 2424242424 252525 26 2727 28 2929 3030 3131 3232 2 3 4 5 6 7 8 9 1112131418202232 3 4 8 1418202231 4 8 9 1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 3334 3334 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4

  9. CS0 and DONOT ISOLATE with CC=0 for round 2 158 7 6 3 4 4 4 5 1 3 1 2 4 1 1 2 2 1 2 1 2 1 4 2 2 2 2 2 4 3 4 5 14 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 E(rd2) 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 17 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 22 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 33 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 34 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 G7 • This 2nd round there will be no CS=0 deletions (since we have nothing but singletons and 3cliques) , • so we could look at CC=0. If we do, we would delete 1&5, 1&6, 1&11, 1&9. 4. Then during the next round of CS=0 deletions, 1&7 and 5&7 will have no common siblings and will delete. Note: In 3. we break 4 3cliques! Should we? Also the remaining white-green connections form 3cliques. Should they be broken? If k-cliques (k3) are not to be preserved, what kind of communities are we going to end up with? By what measure is Fortunato’s white blue green yellow partition considered a good one? (certainly not by any measure which values cliques). S1Prd2 pairwise ANDs 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 0 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 11121314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 34 28303334 2632 32 3034 34 3234 3334 3334 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4

  10. CS0 and DONOT ISOLATE with 3CLIQUE deletion for round 2 158 7 6 3 4 4 4 5 1 3 1 2 4 1 1 2 2 1 2 1 2 1 4 2 2 2 2 2 4 3 4 5 14 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 E(rd2) 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 17 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 22 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 33 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 34 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 Common Siblings and 3cliques Thm: An edge, (h,k) has No Common Siblings i.e., CSh,k ShSk= iff Eh&Ek pure0 iff that edge is not involved in a 3clique. Proof: An edge (g,f) has a common sibling, k, iff (g,f,k) is a 3clique. Thus, removing all edges with NoCommonSiblings leaves only 3cliques (of course, if the DONOT ISOLATE rule is in place, it also leaves the isolates.) g m k • Keep a list of vertices with 1 or 2 remaining siblings (edges they participate in the DO NOT DELETE): 10 12 13 15 16 17 18 19 20 21 22 23 25 26 27 28 29 f G7 2.  S1P rd2 pairwise AND (of vertices of an edge) with count=2, if the two common siblings do not form and edge themselves (and thus, the 4 form a 4vertex 1plex = two 3cliques with a common edge, namely the original pair) delete the edge of that original pair. If count=1, deleted the edge of that original pair. 5 31 11 6 7 9 CS(1,5)={7,11} not an edge, so delete 1,5 CS(9,31)={33,34} not an edge, so delete 9,31 CS(1,11)={5,6} not an edge, so delete 1,11 CS(1,6)={7,11} not an edge, so delete 1,6 CS(3,9)={1,33} not an edge, so delete 3,9 CS(24,30)={33,34} not an edge, so delete 24,30 CS(1,7)={5, 6} not an edge, so delete 1,7 CS(3,33)={9}, so delete 3,33 CS(30,34)={27}, so delete 30,34 CS(1,9)={3}, so delete 1,9 CS(6,7)={17}, so delete 6,7 CS(32,34)={29}, so delete 32,34 S1Prd2 pairwise ANDs 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 0 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 11121314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 34 28303334 2632 32 3034 34 3234 3334 3334 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 That ends Round-2. If we would do a Round-3 of CS=0 again, (29,34) deletes since there are no common siblings. The result is very very close to GN! 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4

  11. CS0 with DONOT ISOLATE rule on G10 CS0 alone separates all 8 colored communities. It may also delete other edges. b7 a3 b8 b6 b5 b9 a5 c3 a4 c2 G10 c4 2 1 3 4 a7 5 a6 a8 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 85 36 40 89 10 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 56 h9 34 59 68 c5 44 79 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 17 46 i0 62 63 54 49 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 g2 73 g3 g6 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 e5 g0 e2 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 e9 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=266. Vertices with OutDeg=0 (leaves) do not have pTrees shown because pTrees display only OutEdges and thus those OD=1 have a pure0 pTree. f0 f1 f5 f2 f4 f3

  12. 1 4 1 1 2 8 4 1 101 1 2 6 8 1 1 1 1 3 2 1 1 1 2 1 1 1 9 1 SkP, k=2,3,4 for vertices 1,2,3,33,34 33 2 2 3 3 3 333 1 1 1 1 1 2 2 2 3 3 3 3 3 3333333333343434341 1 1 1 333 1 2 3 6 9 321 3 141 2 9 28333 9 2430329 1424286 326 32301 6 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Count only Shortest Path Participations emanating from vertices with S1P-counts  50% of the maxS1Pcount=16 (i.e., 8). This specifies starting vertices of 1 2 3 33 34 only 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34SPPC 1 17 31 5 16 5 3 5 5 5 4 4 4 13 2 4 2 1 1 1 3 11 1 1 1 2 3 1 4 3 41 4 1 5 6 11 7 8 9 2 4 10 11 12 13 14 2 15 1 16 1 17 18 19 1 20 21 1 22 23 1 24 2 1 1 1 25 2 5 3 26 5 27 1 28 1 29 30 3 5 31 32 1 33 34 G7

  13. 34 1 34 9 1 1 1 1 1 1 34 34 34 34 9 34 9 1 2 3 6 9 32 r 9 14 24 28 34 r 1 6 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E 34 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Count only Shortest Path Participations emanating from vertices with S1P-counts  75% of the maxS1Pcount=16 (i.e., 12). This specifies starting vertices of 1 34 only 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 1 3 6 1 4 104 4 37 4 4 4 1 4 1 4 4 2 1 1 1 3 1 1 1 3 3 1 1 3 3 1 3 3 3 4 1 1 3 5 1 1 6 1 1 7 7 1 8 9 1 3 47 10 1 11 12 13 14 4 15 1 4 16 1 4 17 18 19 1 4 20 1 21 1 4 22 23 1 4 24 3 1 1 1 6 25 1 3 3 26 3 27 1 4 28 3 29 1 1 30 1 4 31 1 1 32 1 1 33 34 G7

  14. G5 3 2 2 1 2 2 2 2 ct 2 1 1 2 2 ct 3 2 2 1 2 2 2 2 ct 2 1 1 2 2 ct 12 10 2 4 4 2 4 2 ct 0 0 0 0 0 0 0 0 ct 1 2 0 1 1 0 1 0 ct 0 0 0 2 1 0 1 0 ct 0 0 0 0 0 ct 1 2 4 5 7 SP2 1 2 4 5 7 SP 1 2 3 4 5 6 7 8 E 1 2 4 5 7 E 1 2 3 4 5 6 7 8 SP2 1 2 3 4 5 6 7 8 SP 1 2 3 4 5 6 7 8 SP4 1 2 3 4 5 6 7 8 SP3 1 2 3 4 5 6 7 8 SPPC (Shortest Path Participation Counts) 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 3 0 3 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 4 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 1 2 0 0 1 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Delete (1,2) And {3,6,8} and do over. 1 2 3 8 SP gives connectivity comp partition: CC(1)={1,5,7} is a 0plex since EdgeCt=3=COMBO(3,2)-0. CC(2)={2,4} is a 0plex since EdgeCt=1=COMBO(2,2)-0. 4 7 5 6 SP gives connectivity comp partition: CC(1)={1,2,4,5,7} is a 5plex since EdgeCt=5=COMBO(5,2)-5. CC(3)={3,6,8} is a 0plex since EdgeCt=3=COMBO(3,2)-0

  15. 2 1 2 3 2 3 2 1 1 1 1 3 2 0 2 2 E 31 7 31 29 7 27 7 7 3 3 3 9 2 0 2 2 SPPC a e b 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 3 3 3 0 0 0 0 0 8 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 g 0 0 f 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 3 g 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 7 f 0 0 0 0 7 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 6 f 0 0 0 6 0 6 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9 d 1 2 3 4 5 6 7 8 9 a b c d e f g 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 f g b c 3 2 3 1 1 1 1 2 2 2 2 0 0 0 0 0 SP2 SP gives connectivity comp partition: CC(1)={1 2 3 4 5 6 7 8} is a 20plex since EdgeCt=8=COMBO(8,2)-20. CC(9)={9 a b c} is a 3plex since EdgeCt=3=COMBO(4,2)-3 CC(d)={d f g} is a 0plex since EdgeCt=3=COMBO(3,2)-0. CC( e)={e} 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 SP2 all pure0 2 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 SP3 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 5 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 7 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 SP gives connect comps: CC(1)={1}, CC(5)={5 6 7} Is a 0plex since EdgeCt34=COMBO(3,2)-0 Done! b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 7 Delete (1,3) (SPPC=16 max) and delete {d f g}, {e} and do over. Also delete {9 a b c} as a 4VetexHubSpoke3plex. 7 7 7 7 7 7 7 7 3 3 3 3 2 0 2 2 SP 0 1 0 2 1 2 1 1 0 0 0 0 0 0 0 0 SP4 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 2 2 2 SP 0 2 2 2 E 1 1 1 3 2 3 2 1 E 2 2 2 0 1 0 1 2 SP2 3 3 3 3 3 3 3 3 SP 3 3 3 9 9 5 4 3 SPPC (Shortest Path Participation Counts) SP3 all pure0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 0 1 1 1 0 0 0 0 8 0 1 1 0 0 0 0 0 8 0 0 0 1 0 0 0 0 8 0 0 0 3 0 0 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 0 0 0 0 0 3 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 5 0 0 0 0 0 1 1 0 2 0 0 0 3 0 0 0 0 2 0 0 1 1 0 0 0 1 2 0 0 0 1 0 0 0 0 2 0 0 1 0 0 0 0 1 6 1 0 0 0 1 0 1 0 6 1 0 0 0 1 0 1 0 3 0 0 0 3 0 0 0 0 3 0 1 0 1 0 0 0 1 3 0 1 0 0 0 0 0 1 3 0 0 0 1 0 0 0 0 7 0 0 0 0 1 1 0 0 7 0 0 0 0 1 1 0 0 4 0 3 3 0 0 0 0 3 4 0 1 1 0 0 0 0 1 4 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 5 0 0 0 0 0 1 1 0 5 1 0 0 0 0 1 1 0 5 1 0 0 0 0 0 0 0 5 0 0 0 0 0 2 2 0 6 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 6 1 0 0 0 1 0 1 0 6 3 0 0 0 1 0 1 0 7 0 0 0 0 2 2 0 0 7 0 0 0 0 1 1 0 0 7 1 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 SP gives connect comps: CC(1)={1 5 6 7} 2plex EdgeCt=4=COMBO(4,2)-2. CC(2)={2 3 4 8} is a 3plex since Ect=3=COMB(4,2)-3 (a 4VertexHubSpoke) 0 2 0 0 1 0 2 2 0 0 0 0 0 0 0 0 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SP6 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 Delete{2 3 4 8} 4VHubSpoke3plex, (1,6) G6

  16. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E E 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 15 16 16 15 6 6 14 20 21 15 20 26 11 6 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 8 8 5 8 1 10 10 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 8 8 8 8 1 1 10 1 1 8 1 0 1 1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 wt V#> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 SP1 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 -1 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 -1 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 -1 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 -1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 WeightSum 15 -6 -3 -15 -24 -21 -21 -21 -18 -27 -24 -30 -27 -18 -27 -27 -27 -27 -27 -24 Nbrs1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 Nbrs34 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 -20 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 -20 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 If ( WtSum>=-20 & Nbr(1) ) then 1 else 0. Agglomerative, based on weighted sum SPkidentifies 1 and 34 as centers. Then among their individual nbrs, wt V#> 21 22 23 24 25 26 27 28 29 30 31 32 33 34 2 SP1 2 2 2 5 3 3 2 4 3 4 4 6 11 16 -1 SP2 16 16 16 15 6 6 14 20 21 15 20 26 11 6 -1 SP3 6 8 6 4 23 23 6 8 8 5 8 1 10 10 -1 SP4 8 7 8 8 1 1 10 1 1 8 1 0 1 1 -1 SP5 1 0 1 1 0 0 1 0 0 1 0 0 0 0 WeightSum -27 -27 -27 -18 -24 -24 -27 -21 -24 -21 -21 -15 0 15 Nbrs1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 Nbrs34 1 0 1 1 0 0 1 1 1 1 1 1 0 1 -20 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -20 1 0 1 1 0 0 1 1 1 1 1 1 0 1 select their communities with a threshold on the weighted sum (=-20) giving the light green “1community” and black “34community (overlapping). Next, excise those and iterate. When all are in a community probably do a k means reshuffle to improve? Using weights of 0,1,2,4,6 for SP1,2,3,4,5 resp. wt V#> 5 6 7 8 11 12 13 17 18 22 25 26 33 0 SP1 3 4 4 4 3 1 2 2 2 2 3 3 11 1 SP2 14 13 13 17 14 15 14 3 15 16 6 6 11 2 SP3 8 8 8 11 8 9 9 12 8 8 23 23 10 4 SP4 8 8 8 1 8 8 8 8 8 7 1 1 1 6 SP5 0 0 0 0 0 0 0 8 0 0 0 0 0 WeightSum 62 61 61 43 62 65 64 107 63 60 56 56 35 SP1|2(17) 1 1 1 0 1 0 0 1 0 0 0 0 0 60 1 1 1 0 1 0 0 1 0 0 0 0 0 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 Iterate again on the remaining G7 17 SP5 8=5dg Using weights of5,5,1,1,0 for SP1,2,3,4,5 resp. wt V#> 8 12 13 18 22 25 26 33 5 SP1 4 1 2 2 2 3 3 11 5 SP2 17 15 14 15 16 6 6 11 1 SP3 11 9 9 8 8 23 23 10 1 SP4 1 8 8 8 7 1 1 1 0 SP5 0 0 0 0 0 0 0 0 WeightSum 117 97 97 101 105 69 69 121 SP1|2(8) 1 1 1 1 1 0 0 0 SP1|2(33) 0 0 0 0 0 1 1 1 97 1 1 1 1 1 0 0 0 69 0 0 0 0 0 1 1 1 This method uses site betweeness, not edge betweenenss 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 10,25,26,28,29, 31 33,34 not shown (only 17 on, 8 only 27 turned on 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

  17. To extend to PT: kListEh PT2hk=Ek after zeroing the h bit of Ek PTG1 E3 pred=(NPZ)|(PZ&AcyclicPathEnd) APTG1 APTG1 E3predicate = (NPZ&NotCycleEnd)| (PZ&AcyclicPathEnd) PTG1, extension of EG1 E 2-lev stri=|V|=4 APPENDIX SPTG1 Edge, E, Path(PT), ShortestPathv(SPT),AcyclicPath(APT) Trees andCycleList(CL) of G1 G1 E3key 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 PE3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 PE2 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 E2key 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 1111 kListPT2hj PT3hjk=Ek after zeroing Ek j bit. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 APTG1 1 1 1 1 PTG1 1 1 1 1 (pred is NotPureZero) First, construct stride=|V|, 2-level Edge pTree, all others are constructed concurrently from it. 2LEG1 1 1 1 1 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 SPTG1, init E1=SP1,1E2=SP2,1E3=SP3,1 E4=SP4,1 4 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 kListPT3hij PT4hijk=Ek after zeroing i and j bits of Ek E 2levstr=4 pred=NPZ All are 3 hop cycles. Each has 3 start pts , 2 directions. Each repeat 6 times. 6/6=1 3hop cycles (1341) 1 1 0 0 1 1 2 1 0 0 0 1 3 1 1 0 0 1 4 1 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 1 0 0 1 1 1 0 0 1 1 2 0 0 0 1 2 0 0 0 1 3 1 0 0 1 3 1 0 0 1 4 1 1 1 0 4 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 3 4 CLG1 1 0 2 1 1 2 2 0 2 1 3 1 2 0 1 E one-level E 1lev, pred=NPZ 1341 3413 1431 EG1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 PE1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 E1 key 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 key 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 4134 3143 4314 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 1 2 0 1 0 0 2 2 1 0 1 0 3 2 0 1 0 0 1 3 4 0 1 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 1 4 0 1 0 0 1 3 4 1 1 0 0 1 4 3 1 0 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 1 4 0 1 1 0 4 3 1 0 0 0 1 3 4 1 0 0 1 0 4 1 3 0 0 0 1 SPT is completed. For Big Graphs, could stop here (e.g., Friends has ~1B vertices but a diameter of 4, so we would only need to build PT 4-hop paths) and possible expressed as a tree of lists rather than a tree of bitmaps. For sparse BigGraphs, E could be leveled further and/or a tree of lists (then APT, SPT will be also). SPT(G)k (with k turned on) is mask (>0 is “yes”) for connectivity comp, COMP(G)kvk. For bitmap of COMPkbitslicing SPT (SPTk,h..SPTk,0k=1..|V| then COMPk ORj=h..0SPTk,h. SPT structure may be useful as separate “categorical” bitmaps  Shortest Path Length (SPk,h h=1..H. Also keep a mask of Shortest Paths so far, SPSFk vertex, k. With each new SP bitmap, SPB, SPSFkSPSFk| SPB, SPk,h+1 SPB & SPSFk. 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , SPSFk 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 1 4 0 1 1 0 3 1 0 0 0 1 4 1 0 0 1 0 4 3 1 0 0 0 1 3 0 0 0 1 2 4 1 0 1 0 3 4 1 1 0 0 1 3 4 1 1 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 4 1 0 0 1 0 4 1 3 0 0 0 1 1 4 3 1 0 0 0 3 1 4 0 1 1 0 4 3 1 0 0 0 1 SPTgives the Connectivity Component Partition; Maximal Cliques (go across SPk,1 then look within subsets of those k’s for commonality); Note, Cliques are 0-plexes. Each mask, SPk,1 masks a 1-plex. Each SPk,1&SPk,2 masks a 2-plex (which is SPSFk,2? So if we save each SPSF instead of overwriting, we have k-plex masks w/o further work?), etc. Next construct predicates for each Path related data structures, PT APT SPT SPSF, to make them into pTrees on a k-path table, E, E2, E3, … 1 3 4 0 1 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 1 4 0 1 0 0 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E PT SPTAPT of graph as predicate Trees on E(MaxPathLength). 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , SP3,1 SP4,1 SPVertex=3, Len=2 SP3,1|2 SP4,1|2 SP1,1 SP2,1 SP1,2 SP2,2 SP1,1|2 SP2,1|2 2 1 0 0 0 1 3 1 1 0 0 1 4 1 1 1 1 0 1 1 0 0 1 1 2 2 1 0 1 0 3 2 0 1 0 0 2 12 1 0 1 1 3 12 1 1 0 1 4 12 1 1 1 0 1 2 0 1 0 0 1 12 0 1 1 1 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

  18. SG Clique Mining K=2: 2Cliques (2 vertices): 12 13 14 1623 24 34 56 67Find endptsof each edges (Int((n-1)/7)+1, Mod(n-1,7) +1) key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 PE 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 E 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 EU 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 C 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CU 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4 5 6 7 8 9 k=3: 123 124 134 234 k=4: 1234 (123 124 234 are cliques) 123,1341234. 123.2341234. 124,1341234. 124, 2341234. 134,2341234. 1234 only 4-clique Using the EdgeCountthm: on C={1,2,3,4}, CU=C&EU C is a clique since ct(CU)=comb(4, 2)=4!/2!2!=6 k=2: E=12 13 14 16 23 2434 56 57 67. k=3: 123 124 134 234 567 8 EC, requires counting 1’s in mask pTree of each Subgraph (or candidate Clique, if take the time to generate the CCSs – but then clearly the fastest way to finish up is simply to lookup the single bit position in E, i.e., use EC). EdgeCount Algorithm (EC): |PUC| = (k+1)!/(k-1)!2! then CCCS The SG alg only needs Edge Mask pTree, E, and a fast way to find those pairs of subgraphs in CSk that share k-1 vertices (then check E to see if the two different kth vertices are an edge in G. Again this is a standard part of the Apriori ARM algorithm and has therefore been optimized and engineered ad infinitum!) PE(4,8)=1 2348CS4 PE(3,8)=1 1348CS4 PE(4,8)=1 1248CS4 PE(2,6)=0 PE(2,6)=0 6 G3 6 G4 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 8.1 8,2 8,3 8,4 8,5 8,6 8,7 8.8 E 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 6 G2 have PE(2,4)=1 1234CS4 PE(6,7)=1 567CS3 PE(6,7)=1 567CS3 PE(2,4)=1 124CS3 5 5 5 k=4: 1234 1238 1248 1348 2348 k=2: 12 13 14 16 23 2434 56 57 67 18 28 38 48. 7 7 7 PE(2,4)=1 124CS3 PE(1,5)=0 PE(1,5)=0 PE(2,3)=1 123CS3 PE(2,4)=1 1234CS4 PE(4,8)=1 148CS3 PE(2,3)=1 So 123CS3 already have 567 PE(1,7)=0 have PE(1,7)=0 PE(6,8)=0 have Have PE(3,8)=1 238CS3 k=5: 12348 = CS5. PE(2,8)=1 128CS3 PE(3,8)=1 138CS3 2 2 2 1 1 1 PE(2,3)=1 234CS3 PE(3,8)=1 1238CS4 PE(4,8)=1 248CS3 PE(4,8)=1 348CS3 PE(2,3)=1 234CS3 Have 123CS3 have have 124CS3 Have Have 1234 PE(1,4)=1 134CS3 PE(1,4)=1 134CS3 PE(4,8)=1 12348CS5 Have 4 3 4 4 3 3 k=3: 123 124 134 234 567 128 138 148 238 248 348

  19. TheEdgepTree(E), PathTree(PT), ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of a graph, G5 PTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 EG5 2-level str=8 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 3 6 0 0 0 0 0 0 0 1 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 2 1 0 0 0 0 1 0 1 0 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 1 5 7 1 0 0 0 0 0 0 0 1 7 5 1 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 3 6 8 0 0 1 0 0 0 0 0 3 8 6 0 0 1 0 0 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 5 1 7 0 0 0 0 1 0 0 0 5 7 1 0 0 0 0 1 0 0 0 6 3 8 0 0 0 0 0 1 0 0 6 8 3 0 0 0 0 0 1 0 0 7 1 2 0 0 0 1 0 0 0 0 7 1 5 0 0 0 0 0 0 1 0 8 6 3 0 0 0 0 0 0 0 1 8 3 6 0 0 0 0 0 0 0 1 7 5 1 0 1 0 0 0 0 1 0 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 1 2 3 8 CLG5 1571 APTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 1751 4 7 3683 5 6 3863 5175 5715 2 1 0 0 0 0 1 0 1 0 3 6 0 0 0 0 0 0 0 1 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 6386 6836 7157 7517 8368 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 7 1 2 0 0 0 1 0 0 0 0 7 5 1 0 1 0 0 0 0 0 0 8638 PT Clique Miner Algorithm A clique is all cycles Extend to a k-plex(k-core) mining algorithm? PT(=APT+CL), SPT are powerful datamining tools with closure properties (to eliminate branches) . SPTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 2 0 2 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 4 2 1 0 0 3 0 3 0 5 1 2 0 3 0 0 1 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 7 1 2 0 0 1 0 0 0 7 1 2 0 3 1 0 0 0 8 0 0 1 0 0 1 0 0 1 0 1 0 2 1 0 1 0 5 1 2 0 0 0 0 1 0 4 2 1 0 0 0 0 0 0 Max clique MiningA kCycle is a kCliqueiff it’s found in CLk as PERM(k-1,k-1)/2=(k-1)!/2 kCycles (e.g., vertices are repeated in CL for 3cycles, 2!/2=1; 4cycles, 3!/2=3; 5cycles, 4!/2=12; 6cycles, 5!/2=60. 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 Downward closure: Once, a 4cycle 12341 is established as a 4clique (by the fact that {1,2,3,4} occurs 3!/2=3 times in CL), all 3vertex subsets are 3cliques {1,2,3},{1,2,4},{1,3,4}, so no need to check further. k-plex (missing  k edges) mining alg? k-core (has  k edges) mining alg? Density (internal edge density >> external|avg) mining alg? Degree (internal vertex degree >> external|avg) mining alg? DiameterG5 is max{Diameterk} = max{ 2,2,1,3,2,1,3,1}=3. Connected comp containing V1, COMP1={1,2,4,5,7}. Pick 1st vertex not in COMP1,3, COMP3 ={3,6,8}. Done. The partition is { {1,2,4,5,7}, {3,6,8} }. To pick the first vertex not in COMP1, mask off COMP1 with SPTv1’ and then pick the first vertex in this complement.

  20. E=A1Ps 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1 SP1&2 a e b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9 d 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 b f c g A6Ps A5Ps A4Ps A2Ps cycles in blue (not in APT) A3Ps 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g SP1&2&3 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 SP2 A c 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 D g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 G f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b c 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 D f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 F d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 F g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 G d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 4 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 6 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 8 4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 c 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 SP1&2&3&4 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 SP3 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 5 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5 7 6 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 5 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 8 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G F d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 D G f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 7 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 5 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 D F g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 F D g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 F G d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6 5 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 G D f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 7 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 8 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 2 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 4 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 5 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 5 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 5 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 6 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 7 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 7 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 7 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 7 7 7 7 7 7 7 7 7 3 3 3 3 2 0 2 2 SP1&2&3&4&5 COMPLETE 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 SP4 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 1 0 0 0 0 0 0 0 0 7 5 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 4 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SP6 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 G6

  21. 9 13 19 16 13 12 13 17 24 19 14 25 14 25 15 15 3 15 16 26 15 16 16 15 6 6 13 20 21 15 20 26 11 6=2dg 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16=1deg All Shortest Path pTrees for a unipartite undirected graph, G7 (SP1, SP2, SP3, SP4, SP5) 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP1 10,25,26,28,29,33,34 not shown (only 17 on, 1=4dg) 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 G7 17 SP5 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 11 4 11 8 8 8 12 3 11 8 8 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 7 8 5 8 1 10 10=3dg 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 8=5dg 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP3 0 0 0 0 0 0 0 0 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0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  22. Trying Hamming Similarity to detect communities on G7 and G8 1 5 4 2 3 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 Zachary's karate club, a standard benchmark in community detection. (best partition found by optimizing modularity of Newman and Girvan) 41 46 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 =1deg 42 8 45 47 7 9 13 19 16 13 12 13 17 24 19 14 25 14 25 15 15 3 15 16 26 15 16 16 15 6 6 13 20 21 15 20 26 11 6 =2deg 44 6 43 40 8 11 4 11 8 8 8 12 3 11 8 8 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 7 8 5 8 1 10 10 =3deg 9 39 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8 =4deg 38 53 48 12 52 1 1 8 1 1 1 1 1 1 =5deg G8 10 13 To produce an [all?] actual shortest path[s] between x and y: Thm: To produce a [all?]: S2P[s], take a [all?] middle vertex[es], x1, from SP1x & SP1y, produce: xx1y; S3P[s], take a [all?] vertex[es], x1, from SP1x and a [all?] vertex[es], x2, from S2P(x1,y): xx1x2y etc. Is it productive to actually produce (one time) a tree of [all?] shortest paths? I think it is not! Hamming similarity: S(S1,S2)=DegkDif(S1,S2) 14 11 17 1 2 3 4 1 2 3 4 5 6 7 36 54 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 24 35 15 23 22 14 20 20 17 15 16 24 30 27 18 39 28 42 Can see that this Works Poorly At 1. 37 21 49 17 17 25 17 2 2 17 24 18 1 14 3 24 7 7 25 Not working! On the other hand, our standard community mining techniques (for kplexes) worked well on G7. Next slide let’s try Hamming on G8. 19 34 20 27 25 18 50 51 26 30 29 28 33 31 32 G7 3 7 12 3 25 Deg1 4 4 4 4 4 b a 5 6 4 5 g 9 7 4 6 b 2 b 8 6 4 f 9 f 4 9 3 8 6 d 4 5 4 5 4 2 3 6 7 5 7 6 7 3 5 3 5 3 4 9 6 5 19 Deg2 5 8 12 17 8 16 17 16 4 24 21 21 26 20 20 20 19 16 19 23 30 13 15 22 14 20 18 11 14 15 10 15 14 21 14 17 10 4 3 2 4 3 10 21 8 10 15 18 15 15 10 17 18 35

  23. G9, Agglomerative clustering of ESP2 using Hamming Similarity In ESP2, using Hamming similarity, we get three Event clusters, clustering events iffpTrees [Hamming] identical: EventCluster1={1,2,3,4,5} EventCluster2={6,7,8,9} EventCluster3={10,11,12,13,14} ESP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 The Degree % of affiliation of Women with R,G,B events is: RGB 1 100% 75% 0% 2 80% 75% 0% 3 80% 100% 0% 4 80% 75% 0% 5 60% 25% 0% 6 40% 50% 0% 7 20% 75% 0% 8 0% 75% 0% 9 20% 75% 0% 10 0% 75% 20% 11 0% 50% 40% 12 0% 50% 80% 13 0% 75% 80% 14 0% 75% 100% 15 0% 50% 60% 16 0% 50% 0% 17 0% 25% 20% 18 0% 25% 20% W WSP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W WSP3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 6 7 6 6 10 10 10 11 10 10 10 8 7 6 9 12 12 12 8 7 8 8 4 4 4 3 4 4 4 6 7 8 5 2 2 2 3 3 6 4 8 8 10 14 12 5 4 6 3 3 WSP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ESP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 ESP3=ESP1’ and ESP4=ESP2’ so again, in this case, all info is already available in ESP1 and ESP2 (all shortest paths are of length 1 or 2). We don’t need ESPk k>2) G9 9 9 9 9 9 e e e e 9 9 9 9 9 18 16 18 18 12 16 16 17 18 18 17 17 18 18 18 17 13 13 Clustering Women using Degree% RGB affiliation: WomenClusterR={1,2,4,5} WomanClusterG={3,6,7,8,9,10,11,16,17,18} WomanClsuterB={12,13,14,15} WSP3=WSP1’ and WSP4=WSP2’ so, in this case, all information is already available in WSP1 and WSP2 (All shortest paths are of length 1 or 2) (We don’t need WSPk k>2) This clustering seems fairly close to the authors. Other methods are possible and if another method puts event6 with 12345, then everything changes and the result seem even closer to the author’s intent..

  24. K-plex search on G9 (A k-plex is a SG missing  k edges If H is a k-plex and F is a ISG, then F is a kplex A graph (V,E) is a k-plex iff |V|(|V|-1)/2 – |E| k WSP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Events123456789abcde 14*13/2=91 degs=88888dddd88888 |Edge|=66 kplex k25 h f h f b f f g h h g g h h h g c c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Events23456789abcde Not calculating k degs=7777cccc88888 Until it gets lower ESP2 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 0 1 1 1 1 1 1 1 1 0 0 0 0 0 Events3456789abcde14*13/2=91 degs=666bbbb88888 |Edges|=66 kpl Events456789abcde14*13/2=91 degs=55aaaa88888 |Edges|=66 kplex k25 Women123456789abcdefghi 18*17/2=153 degs=hfhfbffghhgghhhgcc |Edges| =139 kplex k14 Events56789abcde14*13/2=91 degs=4999988888 |Edges|=66 kplex k25 Women123456789abcdefgh18*17/2=153 degs=gfgfbfffggffgggfc |Edges| =139 kplex k14 Events6789abcde 9*8/2=36 A 9Clique! degs=888888888 |Edges|=36 kplex k0 Women123456789abcdefg18*17/2=153 degs=ffffbffeffeefffe |Edges| =139 kplex k14 8 8 8 8 8 d d d d 8 8 8 8 8 So take out {6789abcde} and start over. 17 15 17 15 11 15 15 16 17 17 16 16 17 17 17 16 12 12 Women12346789abcdefg 15*14/2=105 degs=eeeeeeeeeeeeeee |Edges| =105 15kplex k0 15Clique Events12345 5*4/2=10 |Edges|=10 kplex k 0 A 5clique! degs: 44444 So take out {12346789abcdefg} and start over. If we had used the full algorithm which pursues each minimum degree tie path, one of them would start by eliminating 14 instead of 1. That will result in the 9Clique 123456789 and the 5Clique abcde. All the other 8 ties would result in one of these two situations. How can we know that ahead of time and avoid all those unproductive minimum degree tie paths? Women5hi 3*2/2=3 degs=011 |Edges| =1 kplex k2 G9 Womenhi 2*1/2=1 degs=11 |Edges| =1 kplex k0 Clique We get no information from applying our kplex search algorithm to WSP2. Again, how could we know this ahead of time to avoid all the work? Possibly by noticing the very high 1-density of the pTrees? (only 28 zeros)? Every ISG of a Clique is a Clique so 6789 and 789 are Cliques (which seems to be the authors intent?) If the goal is to find all maximal Cliques, how do we know that CA=123456789 is maximal? If it weren’t then there would be at least one of abcde which when added to CA=123456789 would results in a 10Clique. Checking a: PCA&Pa would have to have count=9 (It doesn’t! It has count=5) and PCA(a) would have to be 1 (It isn’t. It’s 0). The same is true for bcde. The same type of analysis shows 6789abcde is maximal. I think one can prove that any Clique obtained by our algorithm would be maximal (without the above expensive check), since we start with the whole vertex set and throw out one at a time until we get a clique, so it has to be maximal? The Women associated strongly with the blue EventClique, abgde are {12 13 14 15 16} and associated but loosely are {10 11 17 18}. The Women associated strongly with the green EventClique, 12345 are {1 2 3 4 5} and associated but loosely are {6 7 9}

  25. G10 E=SP1 2level pTrees LevelOneStride=19 (labelled 0-i), Level0Stride=10 (labelled 0-9) Note: SP1 should be called S1PDV for “Shortest 1 Path Destination Verticies, because each one, e.g. S1PDV(v1) maps all such destination verticies from that given starting vertex, v1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 OutDeg 1 8 1 0 0 0 0 1 9 1 0 0 0 0 2 0 1 0 0 0 0 2 1 1 0 0 0 0 2 2 1 0 0 0 0 2 3 1 0 0 0 0 2 4 1 0 0 0 0 2 5 1 0 0 0 0 2 6 1 0 0 0 0 2 7 0 1 0 0 0 2 8 0 1 0 0 0 2 9 0 1 0 0 0 3 0 0 1 0 0 0 3 1 0 1 0 0 0 3 2 0 1 0 0 0 3 3 0 1 0 0 0 3 4 0 1 0 0 0 3 5 0 1 0 0 0 3 6 0 1 0 0 0 3 7 0 0 1 0 0 3 8 0 0 1 1 1 3 9 0 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 b7 1 1 1 1 1 1 1 0 0 0 a3 5 3 4 8 5 4 4 8 5 7 6 9 5 8 7 0 5 9 6 8 6 0 6 7 6 1 6 6 6 3 6 6 6 5 e 7 6 6 6 1 7 1 4 9 b8 tens dig 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 4 5 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 4 7 0 0 0 0 1 0 0 1 0 0 5 1 0 0 0 0 1 0 0 0 0 0 5 2 0 0 0 0 1 1 1 0 0 1 4 6 0 0 0 0 1 1 0 1 0 0 4 8 0 0 0 0 1 4 9 0 0 0 0 1 0 0 1 0 0 5 5 0 0 0 0 1 5 6 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 7 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 3 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 b6 b5 b9 a5 c3 a4 1 1 1 1 0 1 1 0 0 1 c2 c4 2 1 3 4 a7 5 G10 a6 a8 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 units 0 1 2 3 4 5 6 7 8 9 85 36 40 89 10 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 56 h9 34 units 0 1 2 3 4 59 68 c5 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 44 79 0 1 0 0 0 1 1 0 0 0 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 units 0 1 2 3 4 5 6 7 8 9 17 46 i0 62 63 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 54 0 1 0 0 1 49 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 units 0 1 2 3 4 g2 73 g3 g6 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 units 0 1 2 3 4 e5 g0 e2 0 1 0 0 0 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 units 0 1 2 3 4 5 6 7 8 9 e9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=266. Vertices with OutDeg=0 (leaves) do not have pTrees shown because pTrees display only OutEdges and thus those OD=1 have a pure0 pTree. 0 0 0 0 0 0 0 1 0 0 f0 f1 f5 f2 f4 f3

  26. 7OD G10 E=SP1 2level pTrees LevelOneStride=19 (labelled 0-i), Level0Stride=10 (labelled 0-9) 9 OD L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i C 4 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i H 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 211 1 1 1 OutDeg 1 1 2 OD B 5 C 4 B 6 B 7 1 8 1 0 0 0 0 1 9 1 0 0 0 0 2 0 1 0 0 0 0 2 1 1 0 0 0 0 2 2 1 0 0 0 0 2 3 1 0 0 0 0 2 4 1 0 0 0 0 2 5 1 0 0 0 0 2 6 1 0 0 0 0 2 7 0 1 0 0 0 2 8 0 1 0 0 0 2 9 0 1 0 0 0 3 0 0 1 0 0 0 3 1 0 1 0 0 0 3 2 0 1 0 0 0 3 3 0 1 0 0 0 3 4 0 1 0 0 0 3 5 0 1 0 0 0 3 6 0 1 0 0 0 3 7 0 0 1 0 0 3 8 0 0 1 1 1 3 9 0 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 B 4 C 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 7 5 7 7 7 6 7 7 7 7 7 6 h 5 L1 0 1 2 3 4 5 6 7 8 9 a b . . . L0 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 C 0 B 2 C 4 C 1 7 7 C 4 C 2 B 9 C 4 C 3 B 8 5 5 2 8 20 4 OD L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 7 8 0 0 0 0 1 0 0 0 1 0 7 9 0 0 0 0 1 0 0 0 1 1 8 0 0 0 0 1 0 0 0 0 1 0 8 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 9 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 9 1 4 0 4 6 7 6 H 8 H 9 B 1 B 4 C 6 C 7 A 7 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 8 6 8 0 8 7 7 9 8 9 8 5 9 0 A 6 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 1 1 1 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 OD L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 5 1 4 6 5 3 4 8 5 4 4 8 5 5 4 9 5 7 6 9 5 8 7 0 5 9 6 8 6 0 6 7 6 1 6 6 6 3 6 6 6 5 e 7 6 6 6 1 7 1 4 9 9 2 9 1 9 3 9 1 9 5 7 9 9 6 7 8 6 6 2 3 3 3 17 3 2 2 2 2 OD L0 0 1 2 3 4 5 6 7 8 9 4 6 0 0 0 0 1 1 0 1 0 0 4 8 0 0 0 0 1 4 9 0 0 0 0 1 0 0 1 0 0 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 4 5 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 4 7 0 0 0 0 1 0 0 1 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 5 2 0 0 0 0 1 1 1 0 0 1 5 6 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 7 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 3 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 C 5 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 9 7 A 7 9 8 9 1 9 9 8 8 A 0 A 8 1 1 1 1 0 1 1 0 0 1 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 A 1 A 9 A 2 B 0 A 4 A 7 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 8 8 7 8 9 1 9 4 9 1 5 2 A 5 A 3 A 7 20 OD L0 0 1 2 3 4 5 6 7 8 9 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i D 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 H 4 D 2 4 6 7 2 7 3 7 4 C 6 C 7 C 8 C 9 C 6 9 1 D 2 C 7 9 1 D 2 C 8 7 9 D 2 C 9 7 8 D 0 H 4 D 2 D 1 4 5 7 8 D 2 D 4 D 2 D 3 D 2 4 6 8 1 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 L0 0 1 2 3 4 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 D 5 D 2 D 6 F 5 D 2 D 7 D 9 D 2 D 8 F 4 D 2 D 9 9 1 D 2 E 0 9 1 D 2 E 1 7 9 D 2 E 2 D 2 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 1 0 1 0 0 1 1 1 2 1 OD E 3 F 0 D 2 E 4 E 9 D 2 E 5 E 8 D 2 E 6 E 7 D 2 L0 0 1 2 3 4 B 1 B 2 B 2 7 6 h 1 B 3 B 2 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 0 0 0 L0 0 1 2 3 4 0 1 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 H 6 F 7 G 1 G 2 G 4 G 5 G 3 I 0 H 5 H 3 G 6 H 9 H 2 F 6 H 0 H 1 H 8 G 7 G10 leaves (OutDegree=0): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41 44 62 64 67 68 69 70 82 83 84 85 a3 a6 a8 a9 b0 B7 b8 b9 e7 e8 e9 f0 f1 f2 f3 f4 f5 f8 f9 g0 g8 g9 h7 G 6 H 1 F 8 F 7 7 7 H 4 G 0 F 6 G 1 H 4 H 0 H 4 I 0 G 1 H 4 G 7 9 1 H 4 4 4 H 4 H 9 I 0 4 3 7 8 7 9 F 9 F 7 H 4 H 4

  27. C640 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 E040 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 D9 E1 E2 E3 E4 E5 E6 H8 H9 D246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 18 1 G10 E=SP1 Lists 7577 SP2 Lists 8021 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 78 79 83 84 85 7677 19 2 36 2 7776 H5 37 3 6 20 3 38 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 47 48 50 51 71 74 78 B2 D1 H7 7841 43 46 81 88 21 4 D3D2 22 5 7946 81 87 95 96 D4D2 23 6 8141 43 46 81 87 88 95 96 8038 81 24 7 D5D2 E146 72 73 74 81 87 95 96 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E2 E3 E4 E5 E6 C740 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 8178 79 83 84 85 25 8 8638 81 D6E5 D2 39 12 26 9 8680 8746 81 95 96 40 10 27 10 D7D9 D2 8840 41 43 46 76 81 86 88 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 8779 42 46 51 78 B2 D1 H7 28 11 D8E4 D2 8878 91 29 12 E246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E3 E4 E5 E6 D991 D2 30 13 43 41 46 81 88 8985 31 14 45 41 43 45 47 48 50 76 D2 H1 81 88 E091 D2 90A6 C846 72 73 74 81 87 95 96 C6 C7 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 32 15 9127 43 45 47 48 50 51 77 78 79 80 83 84 85 88 91 A6 A7 A8 A9 B0 B2 C4 D2 H4 I0 9140 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 E179 D2 E346 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E4 E5 E6 33 16 34 17 E2D2 46 46 49 55 72 78 B2 D1 H6 H7 35 18 E3F0 D2 36 19 9291 C941 43 46 81 88 E446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E5 E6 E4E9 D2 37 20 23 47 46 49 D2 D046 72 73 74 81 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H0 H1 H2 H3 H4 H5 H6 H7 H8 9240 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 38 21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 9391 48 45 48 50 51 72 74 E5E8 D2 9491 52 E6E7 D2 49 45 47 48 49 50 51 D2 D546 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E6 9579 H3 G1 H8 H6 H5 H2 F6 H1 I0 H0 G6 H3 G5 H4 H4 G5 I0 F7 G1 H9 G2 G3 H8 G4 H6 G6 H9 G7 H0 H5 H1 H2 43 46 78 79 81 H0 H1 H2 H3 H5 H7 H8 40 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H0 H1 H2 H3 H5 H6 H7 H9 F8 G1 H4 G1 H4 F6 F7 G7 G0 F9 H1 G6 I0 H0 H4 41 44 46 81 87 88 95 96 H4 F7 H4 46 81 G1 H0 H1 H3 H4 H5 H6 H7 H8 46 81 H0 H2 H3 H4 H5 H6 H7 H8 44 H4 H9 H4 46 81 H0 H1 H2 H3 H5 H7 H8 77 H4 46 76 81 H0 H1 H2 H3 H6 H7 H8 G6 45 47 48 50 51 77 91 G1 I0 46 81 H0 H1 H2 H3 H5 H6 H7 H8 H4 46 81 H0 H1 H2 H3 H4 H5 H6 H7 H8 46 81 H1 H2 H3 H4 H5 H6 H7 H8 44 H4 H9 91 H4 I0 43 78 79 9340 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 9678 50 45 47 48 49 50 H4 39 29 D141 43 46 72 73 74 81 88 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 97A7 40 27 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 42 45 9891 51 45 47 48 50 9440 46 53 54 55 56 57 58 59 60 61 62 63 64 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 43 78 9988 52 45 47 48 49 50 51 66 67 68 69 70 75 B3 45 46 51 78 B2 D1 H7 D245 47 48 49 78 79 91 E7 E8 E9 F0 A0A8 A1A9 46 45 47 48 50 51 53 46 47 49 D346 72 73 74 C6 C7 C8 C9 D1 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 A2B0 47 48 72 54 46 47 49 A4A7 48 46 47 49 9546 81 87 96 55 46 71 74 A5A3 A7 49 46 71 74 9641 43 46 81 88 56 46 51 55 77 B2 H6 D446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 A798 99 A4 A5 9798 99 A4 A5 50 46 51 55 H6 B1 B2 9840 46 76 81 86 89 90 97 99M A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 63 1 51 46 B276 H1 52 46 53 54 55 56 57 58 59 60 61 62 63 64 49 46 74 D546 72 73 74 C6 C7 C8 C9 D1 D3 D8 D9 E0 E1 E2 E3 E4 E5 E6 B3B2 72 46 48 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4C4 9978 91 53 48 A498 99 A5 B5C4 54 48 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 E8 A598 99 A4 B6B7 55 49 A788 91 A3 73 46 47 49 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B1 76 H1 C0B2 C4 56 50 75 B3 B277 G1 H4 C177 C4 57 69 B3 76 H1 D746 72 73 74 91 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4B5 B6 C0 C1 C2 C3 C2B9 C4 58 70 C3B8 74 46 49 71 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 59 68 B5B4 B6 C0 C1 C2 C3 C4B4 B5 B6 C0 C1 C2 C3 60 67 C076 B4 B5 B6 C1 C2 C3 H1 61 66 D846 72 73 74 C6 C7 C8 C9 D1 D3 D5 D9 E0 E1 E2 E3 E4 E5 E6 E9 c5 45 D5 OD=0: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41 44 62 64 67 68 69 70 82 83 84 85 a3 a6 a8 a9 B0 B7 b8 b9 e7 e8 e9 f0 f1 f2 f3 f4 f5 f8 f9 g0 g8 g9 h7 63 66 C177 B4 B5 B6 C0 C1 C2 C3 H5 7576 H5 C691 D2 65 E7 7676 H5 C2B4 B5 B6 C0 C1 C3 D940 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 E0 E1 E2 E3 E4 E5 E6 H8 H9 77H4 C791 D2 66 61 7845 47 48 50 51 78 79 83 85 91 71 49 C879 D2 C477 B2 B7 B8 B9 C4 C978 72 47 D2 7946 47 48 50 51 78 79 83 84 85 C5 46 51 78 B2 D1 D2 H7 73 48 D2 D0H4 D2 74 49 D2 D178 D2

  28. 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 * 21 * 22 * 23 * 24 * 25 * 26 * 27 * * 28 * 29 ** 30 * 31 * 32 * 33 * 34 * 35 * 36 * 37 38 * 39 40 * * 41 * * 42 * 43 * * * * 44 * 45 * * * * * 46 * * ***** ** * * 47 * * * 48 ** ** * 49 * * * * * 50 * * 51 ** * 52 * 53 * 54 * 55 * * 56 * 57 * 58 * 59 * 60 * 61 * * 62 * 63 * 64 * 65 66 * * 67 * 68 * 69 * 70 * 71 * 72 * * 73 * 74 * * 75 * * 76 * * * * 77 ** * * 78 * * * * * * * 79 * * * * * 80 * 81 *** * * 82 83 * 84 * 85 * * 86 * 87 * 88 * *** * 89 * 90 * 91 * * ** ** * 92 93 94 95 * 96 * 97 * 98 * * 99 * * 100 * 101 * 102 * 103 * 104 * * 105 * 106 * 107 * ** 108 * 109 * 110 * 111 * 112 * * * * 113 * 114 * * 115 * 116 * 117 * 118 * 119 * 120 * 121 * 122 * 123 * 124 ** *** 125 126 * * 127 * * 128 * 129 * 130 131 * * 132 *** *** ** ************** 133 * 134 135 * * 136 137 138 * 139 * * 140 * 141 * 142 * 143 * 144 * * 145 * * 146 * 147 * * 148 * 149 * 150 * 151 152 153 154 155 156 * 157 ** 158 * 159 * 160 * 161 * ** 162 163 164 165 166 * 167 * 168 169 170 * 171 * * 172 * 173 * 174 * * *** ** * * 175 * * 176 * * 177 * * 178 * * 179 * * 180 * * G10 Edge Matrix Raster ordering EM gives the E table cardinality(E) = 180*180 = 32,400. 111111111111111111111111111111111111111111111111111111111111111111111111111111111 111111111122222222223333333333444444444455555555556666666666777777777788888888889999999999000000000011111111112222222222333333333344444444445555555555666666666677777777778 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890

  29. Use both a fore and aft pTree. GN: Remove edge with largest betweeness. Recalc betweenesses; Repeat. 4 3 2 1 S P P C 0 5 0 0 0 5 0 5 4 5 0 5 0 4 0 0 4 4 0 4 0 5 0 4 0 G1_3 5 1 0 1 0 0 0 1 2 0 0 1 0 1 2 3 0 0 0 1 0 1 4 0 0 0 0 1 1 E 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 Ekey 1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5 1 1 0 0 0 0 2 5 0 0 0 1 0 1 0 0 0 0 0 0 1 2 0 0 1 0 1 2 2 1 0 0 0 1 2 3 0 0 0 1 0 1 1 0 1 0 0 0 3 4 0 0 0 0 1 1 1 0 0 1 0 0 4 5 0 1 0 0 0 1 0 0 0 0 0 0 1 2 5 0 0 0 1 0 1 0 0 0 0 0 0 1 2 5 0 0 0 1 0 1 0 0 0 0 0 0 1 2 3 0 0 0 1 0 1 To construct SPPC(hk) =SPPC(kh) (Shortest Path Partic Count) if (hk)E ct 1 + CtS2P(hk) + CtS2P(kh) + CtS3P(hkg) + CtS3P(ghk), g + CtS4P(hkfm) + CtS4P(fhkm) + CtS4P(fmhk) f,m. Etc.

  30. S2P(h) = blue and orange MCFC: Delete the edge(s) with the Minimum # of Common First Cousins, where CFC(h,k)S2P(h) & S2P(k) S2P 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 c d e b a h k f g j i h S2P(k) = red and green G7 S2P All Paths 1 4 1 1 8 4 1 101 1 2 6 9 1 4 1 131 1313121 4 143 7 1 9 11131 1514121 4 8 106 106 2 1 141 106 141 11106 141 106 2 1 1 7 4 1 2 3 4 2 2 139 2 1 8 9 2 102 9 4 8 8 4 151 1 7 2 8 1 1 1 1 3 2 1 1 1 1 1 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 6 7 8 8 8 9 9 9 9 1010111112131414141415151616171718181919202020212122222323242424242425252526262727282828282929293030303131313232323232333333333334343434 2 3 6 9 1 3 141 2 9 28331 2 3 141 7 1 1 1 2 3 1 3 33343 341 6 1 1 1 2 3 34333433346 7 1 2 33341 2 3433341 2 33342628303334262832243230343 2425343 32342433342 33341 252633343 9 2430329 142428 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

  31. Divisive Graph Clustering: Girvan and Neuman delete edges with max “betweenness”, i.e., max participation in shortest paths (of all lengths) Girvan and Newman (Girvan and Newman,02; 04). Edges deleted based on a measures of edge betweeness:. 1. Computation of the edge betweeness for all edges; 2. Removal of edge with largest betweeness: in case of ties with other edges, one is picked at random; 3. Recalculation of betweeness on the running graph; 4. Iteration of the cycle from step 2. We look for situations where pTrees give us an advantage. Can SPPC (Shortest Path Participation Count) be constructed with pTrees more efficiently? What other measure can pTrees make much more efficiently that can help choose the best edge to delete? Later we will try finding the edge with maximum “Fore-Aft” Shortest Path Participation Difference in S1P, S2P, S3P,… (or some combination). pTrees should provide great advantage in the calculation of FAD(h,k). The other important question to answer is: Does it create a good clsutering? key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 SP2 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 E 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 SPPC 0 4 4 4 0 c 0 4 0 1 1 0 0 0 4 1 0 1 0 0 0 4 1 1 0 0 0 0 0 0 0 0 0 5 0 c 0 0 0 5 0 5 0 0 0 0 0 5 0 SP3 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 While constructing Shortest Path pTrees, SP2…, record Shortest Path Participation Count of each edge (SPPC) The edge(s) with max SPPC should be the best candidates for removal? 4 3 3 3 1 3 1 ct 3 3 3 3 1 2 1 ct We will try FAD(h,k)  |S1P(h)&S1P(k)| / |S1P(h)|*|S1P(k)| Or use S2P? Or both? Or S3P? 1 2 3 4 5 6 7 E 1 2 3 4 5 6 7 E 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 2 3 4 5 6 7 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 1 1 1 2 3 2 ct 0 0 0 0 1 0 1 ct 1 2 3 4 5 6 7 SP2 1 2 3 4 5 6 7 SP2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 6 G2 5 0 2 2 2 3 0 3 ct 0 0 0 0 0 0 0 ct 7 1 2 3 4 5 6 7 SP3 1 2 3 4 5 6 7 SP3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 1 4 3 3 3 1 3 1 ct 3 3 3 3 2 2 2 ct 1 2 3 4 5 6 7 SP=SP1 | SP2 | SP3 1 2 3 4 5 6 7 SP 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 2 3 4 5 6 7 SP gives connectivity comp partition: CC(1) = {1}List(SP(1) = {1,2,3,4,5,6,7} is a 12plex since EdgeCt=9=COMBO(7,2)-12 4 3 3 3 3 3 2 2 2 ct 24 6 6 6 5 22 5 ct 1 2 3 4 5 6 7 SPPC 1 2 3 4 5 6 7 SPPC 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 4 0 1 1 0 0 0 4 1 0 1 0 0 0 4 1 1 0 0 0 0 0 0 0 0 0 5 0 c 0 0 0 5 0 5 0 0 0 0 0 5 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 1 1 1 0 0 0 0 4 4 4 0 c 0 SP gives the connectivity component partition: CC(1)={1,2,3,4} 0plex since EdgeCt=12= 2*COMBO(4,2) CC(5)={5,6,7} 1plex since EdgeCt=4=2*(COMBO(3,2)-1) Delete (1,6) and do over.

  32. 3 2 P 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 G1 Divisive Graph Clustering SPPC 0 4 0 0 0 4 0 4 4 4 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 2Pkey 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 2 2 4 4 4 4 4 4 SPPC 0 5 0 0 0 5 0 5 4 5 0 5 0 4 0 0 4 4 0 4 0 5 0 4 0 3 3 2 2 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 G1_1 1 1 1 1 1 1 G1_2 G1_2 G1_3 G1_3 G1_3 G1_3 G1_3 G1_3 GN Delete max SPPC edge. Recalc SPPCs. Repeat. 4 4 5 5 5 5 5 5 5 5 null S 1 P 1 0 1 0 0 0 1 S 1 P 1 0 1 0 0 0 1 S 1 P 5 0 1 0 1 0 2 S 1 P 5 0 1 0 0 0 1 S 1 P 2 1 0 1 0 1 3 S 1 P 2 1 0 1 1 1 4 S 1 P 3 0 1 0 1 0 2 S 1 P 3 0 1 0 0 0 1 S 1 P 4 0 0 1 0 1 2 S 1 P 4 0 1 0 0 0 1 SPPC 0 0 3 2 0 0 0 3 3 0 0 4 1 3 4 0 null S 1 P 1 0 0 1 0 1 S 1 P 1 0 0 1 1 2 S 1 P 2 0 0 0 1 1 S 1 P 2 0 0 0 1 1 S 1 P 3 1 0 0 1 2 S 1 P 3 1 0 0 1 2 S 1 P 4 1 1 1 0 3 S 1 P 4 0 1 1 0 2 3 3 4 4 null null nul SPPC 0 0 1 2 0 0 0 3 1 0 0 2 2 3 2 0 null E 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 E 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 E 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 E 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 E 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 Ekey 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Ekey 1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5 Ekey 1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5 Ekey 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 Ekey 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 null null nul nul nul S 2 P 1 3 0 0 0 1 1 S 2 P 2 4 0 0 1 0 1 S 2 P 3 4 0 1 0 0 1 S 2 P 4 3 1 0 0 0 1 S 2 P 5 2 1 0 1 1 0 3 S 2 P 1 2 0 0 1 1 1 3 S 2 P 3 2 1 0 0 1 1 3 S 2 P 4 2 1 0 1 0 1 3 S 2 P 3 2 1 0 0 0 1 2 S 2 P 3 4 0 0 0 0 1 1 S 2 P 4 3 0 1 0 0 0 1 S 2 P 4 5 0 1 0 0 0 1 S 2 P 5 2 1 0 1 0 0 2 S 2 P 5 4 0 0 1 0 0 1 S 2 P 1 2 0 0 1 0 1 2 S 2 P 2 3 0 0 0 1 0 1 S 2 P 2 5 0 0 0 1 0 1 null Check SPPC(34)=SPPC(43) (verify SPs backwards from hk get counted.) (34)E so ct=1 + CountS2P(34)=1 + CountS2P(43)=1 so ct=3 + CtS3P(34g)=0 + CtS3P(g34)=1, g=1 ct=4 GN says delete (3,4)! nul GN says delete any edge! nul nul nul S 2 P 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 P 1 4 0 1 0 0 1 S 2 P 2 4 1 0 1 0 2 S 2 P 3 4 0 1 0 0 1 nul nul nul nul nul nul nul nul nul S 3 P 1 3 4 0 1 0 0 S 3 P 2 4 3 1 0 0 0 null null null S 3 P 1 2 3 0 0 0 1 0 1 S 3 P 1 2 5 0 0 0 1 0 1 S 3 P 4 3 2 1 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 1 nul GN says delete 12 | 25 | 34 | 36 null null To construct SPPC(hk) =SPPC(kh) (Shortest Path Participation Count) if (hk)E count 1 + OneCountS2P(hk) + OneCountS2P(kh) + OneCountS3P(hkg) + OneCountS3P(ghk), g + OneCountS4P(hkfm) + OneCountS4P(fhkm) + OneCountS4P(fmhk) f,m. Etc. GN: delete 12 | 23 | 25 not 34, 45 6 6 6 6 6 6 6 6 S 1 P 1 0 1 0 0 0 1 2 S 1 P 5 0 1 0 1 0 0 2 S 1 P 6 1 0 1 0 0 0 2 S 1 P 2 1 0 1 0 1 0 3 S 1 P 3 0 1 0 1 0 1 3 S 1 P 4 0 0 1 0 1 0 2 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 5 5 5 5 5 5 5 5 not 23, 16, 45 SPPC 0 7 0 0 5 7 0 6 4 7 0 6 0 7 0 7 0 4 7 0 5 0 7 0 5 5 7 0 S 2 P 1 6 0 0 1 0 0 0 1 S 2 P 2 1 0 0 0 0 0 1 1 S 2 P 3 2 1 0 0 0 1 0 2 S 2 P 3 4 0 0 0 0 1 0 1 S 2 P 3 6 1 0 0 0 0 0 1 S 2 P 4 3 0 1 0 0 0 1 2 S 2 P 4 5 0 1 0 0 0 0 1 S 2 P 5 2 1 0 1 0 0 0 2 S 2 P 5 4 0 0 1 0 0 0 1 S 2 P 6 1 0 1 0 0 0 0 1 S 2 P 1 2 0 0 1 0 1 0 2 S 2 P 2 3 0 0 0 1 0 1 2 S 2 P 2 5 0 0 0 1 0 0 1 S 2 P 6 3 0 1 0 1 0 0 2 SPPC recalculation and repeat steps? Anyone see a shortcut? Or do we just start the calculation over on the reduced graph? Do the pointers help? Since in S2P(hk) one has to search out S2P(kh) and in S3P(hk) one has to find all S3P(hkg) snf D3P(ghk) g In the appendix I begin work on uniquely representing shortest k paths using both a fore and aft pTree. Consider that in G1_4 S3P(16)=2. • Notes: • If any OneCount=0, no subsequence exist. • It might be useful to use ptrs to make this proc easier. • GN edge betweenness specifies pruning (2,4) S 3 P 1 2 3 0 0 0 1 0 0 1 S 3 P 1 2 5 0 0 0 1 0 0 1 S 3 P 1 6 3 0 0 0 1 0 0 1 S 3 P 4 3 2 1 0 0 0 0 0 1 S 3 P 4 3 6 1 0 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 0 1 S 3 P 6 1 2 0 0 0 0 1 0 1 S 3 P 6 3 2 0 0 0 0 1 0 1 S 3 P 6 3 4 0 0 0 0 1 0 1 S 3 P 5 2 1 0 0 0 0 0 1 1 S 3 P 5 2 3 0 0 0 0 0 1 1 S 3 P 5 4 3 0 0 0 0 0 1 1

  33. McS0:“McS0 only with the DONOT ISOLATE rule” round 2. S1P 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G7 Next round the minimum is one. Note that we no longer preserve cliques when the minimum is one. Next round 3 9 have no common siblings and will delete. S1P pairwise ANDs 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 0 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 24242424 2525 26 2727 28 2929 3030 3131 3232 2 3 4 5 6 7 8 9 11121314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 34 28303334 2632 32 3034 34 3234 3334 3334 3334 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4

  34. S1P pairwise ANDs 1 2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 24242424 2525 26 2727 28 2929 3030 3131 32 35 36 37 38 38 37 7 8 9 11121314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 34 28303334 2632 32 3034 34 3234 3334 3334 34 1 b7 a3 b8 b6 b5 b9 a5 c3 a4 c2 c4 2 1 3 4 a7 5 a6 a8 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 85 36 40 89 10 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 56 h9 34 59 68 c5 44 79 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 17 46 i0 62 63 54 49 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 g2 73 g3 g6 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 e5 g0 e2 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=266. Vertices with OutDeg=0 (leaves) do not have pTrees shown because pTrees display only OutEdges and thus those OD=1 have a pure0 pTree. e9 f0 f1 f5 f2 f4 f3

  35. G1 Divisive Graph Clustering Delete edge with zero Common Siblings co-participation. Calculating CS(h,k) is fast with pTrees, but is the resulting clustering a good one? CS0 sats all edges are equal (correct?). 2 2 4 3 3 2 2 3 2 1 1 1 1 G1_1 1 Define CS2(1,2) = S1P(1) & S1P(2) | S1P(1) & S2P(2) | S2P(1) & S1P(2) | S2P(1) & S2P(2), S2P(h)=ORkS2P(hk) G1_2 G1_2 S 1 P 1 & 2 0 0 0 0 0 0 S 1 P 2 & 5 0 0 0 0 0 0 S 1 P 2 & 3 0 0 0 0 0 0 S 1 P 3 & 3 0 0 0 0 0 0 S 1 P 4 & 5 0 0 0 0 0 0 G1_3 4 4 5 5 5 S 1 P 1 & 2 0 0 0 0 0 0 S 1 P 1 & 3 0 0 0 0 0 S 1 P 2 & 4 0 0 0 0 0 S 1 P 2 & 4 0 0 0 0 0 0 S 1 P 3 & 4 1 0 0 0 0 S 1 P 2 & 3 0 0 0 0 0 0 S 1 P 2 & 5 0 0 0 0 0 0 S 1 P 1 0 1 0 0 0 1 S 1 P 1 0 1 0 0 0 1 S 1 P 5 0 1 0 1 0 2 S 1 P 5 0 1 0 0 0 1 S 1 P 2 1 0 1 1 1 4 S 1 P 2 1 0 1 0 1 3 S 1 P 3 0 1 0 0 0 1 S 1 P 3 0 1 0 1 0 2 S 1 P 4 0 0 1 0 1 2 S 1 P 4 0 1 0 0 0 1 S 1 P 1 0 0 1 1 2 S 1 P 1 0 0 1 0 1 S 1 P 2 0 0 0 1 1 S 1 P 2 0 0 0 1 1 S 1 P 3 1 0 0 1 2 S 1 P 3 1 0 0 1 2 S 1 P 4 1 1 1 0 3 S 1 P 4 0 1 1 0 2 3 3 4 4 S 1 P 1 & 3 0 0 0 1 1 S 1 P 2 & 4 0 0 0 0 0 S 1 P 1 & 4 0 0 1 0 1 S 1 P 3 & 4 1 0 0 0 1 CS0 says all edges are equal (seems correct). S 2 P 3 2 1 0 0 0 1 2 S 2 P 3 4 0 0 0 0 1 1 S 2 P 4 3 0 1 0 0 0 1 S 2 P 4 5 0 1 0 0 0 1 S 2 P 5 2 1 0 1 0 0 2 S 2 P 5 4 0 0 1 0 0 1 S 2 P 1 2 0 0 1 0 1 2 S 2 P 2 3 0 0 0 1 0 1 S 2 P 2 5 0 0 0 1 0 1 F A 2 1 2 0 0 1 0 1 2 F A 2 2 3 1 0 0 1 1 3 F A 2 2 5 1 0 1 1 0 3 F A 2 3 4 0 1 0 0 1 2 F A 2 4 5 0 1 1 0 0 2 S 2 P 1 4 0 1 0 0 1 S 2 P 2 4 1 0 1 0 2 S 2 P 3 4 0 1 0 0 1 CS0 says all edges are equal (seems correct). S 2 P 5 2 1 0 1 1 0 3 S 2 P 1 2 0 0 1 1 1 3 S 2 P 3 2 1 0 0 1 1 3 S 2 P 4 2 1 0 1 0 1 3 CS2 says 12 34 45 are the best to delete (more sensitive!) S 3 P 1 2 3 0 0 0 1 0 1 S 3 P 1 2 5 0 0 0 1 0 1 S 3 P 4 3 2 1 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 1 CS0 picks 24. Correct. F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 5 0 0 0 0 0 0 0 F A 1 1 2 0 0 0 0 0 0 0 F A 1 1 6 0 0 0 0 0 0 0 F A 1 3 4 0 0 0 0 0 0 0 F A 1 3 6 0 0 0 0 0 0 0 F A 1 4 5 0 0 0 0 0 0 0 6 6 S 1 P 1 0 1 0 0 0 1 2 S 1 P 1 0 1 0 0 0 1 2 S 1 P 5 0 1 0 1 0 0 2 S 1 P 5 0 0 1 1 0 0 2 S 1 P 6 1 0 1 0 0 0 2 S 1 P 6 1 1 0 0 0 0 2 S 1 P 2 1 0 1 0 0 1 3 S 1 P 2 1 0 1 0 1 0 3 S 1 P 3 0 1 0 1 1 0 3 S 1 P 3 0 1 0 1 0 1 3 S 1 P 4 0 0 1 0 1 0 2 S 1 P 4 0 0 1 0 1 0 2 F A 1 2 3 0 0 0 0 0 0 0 F A 1 2 6 1 0 0 0 0 0 1 F A 1 1 2 0 0 0 0 0 1 1 F A 1 1 6 0 1 0 0 0 0 1 F A 1 3 4 0 0 0 0 1 0 1 F A 1 3 5 0 0 0 1 0 0 1 F A 1 4 5 0 0 1 0 0 0 1 CS0 picks 23, correctly 4 4 3 3 2 2 1 1 G1_5 G1_4 5 5 S 2 P 1 6 0 0 1 0 0 0 1 S 2 P 2 1 0 0 0 0 0 1 1 S 2 P 3 2 1 0 0 0 1 0 2 S 2 P 3 4 0 0 0 0 1 0 1 S 2 P 3 6 1 0 0 0 0 0 1 S 2 P 4 3 0 1 0 0 0 1 2 S 2 P 4 5 0 1 0 0 0 0 1 S 2 P 5 2 1 0 1 0 0 0 2 S 2 P 5 4 0 0 1 0 0 0 1 S 2 P 6 1 0 1 0 0 0 0 1 S 2 P 1 2 0 0 1 0 1 0 2 S 2 P 2 3 0 0 0 1 0 1 2 S 2 P 2 5 0 0 0 1 0 0 1 S 2 P 6 3 0 1 0 1 0 0 2 F A 2 1 2 0 0 1 0 1 1 3 F A 2 1 6 0 1 1 0 0 0 2 F A 2 2 3 1 0 0 1 1 1 4 F A 2 2 5 1 0 1 1 0 0 3 F A 2 3 4 0 1 0 0 1 1 3 F A 2 3 6 1 1 0 1 0 0 3 F A 2 4 5 0 1 1 0 0 0 2 F A 2 1 2 0 0 1 0 0 1 2 F A 2 1 6 0 1 0 0 0 0 1 F A 2 2 3 0 0 0 0 0 0 4 F A 2 2 6 1 0 1 1 0 0 3 S 2 P 1 6 0 1 0 0 0 0 1 S 2 P 2 1 0 0 0 0 0 1 1 S 2 P 3 2 1 0 0 0 0 1 2 S 2 P 3 4 0 0 0 0 1 0 1 S 2 P 3 5 0 0 0 1 0 0 1 S 2 P 4 3 0 1 0 0 1 0 2 S 2 P 4 5 0 0 1 0 0 0 1 S 2 P 5 3 0 1 0 1 0 0 2 S 2 P 5 4 0 0 1 0 0 0 1 S 2 P 6 1 0 1 0 0 0 0 1 S 2 P 1 2 0 0 1 0 0 1 2 S 2 P 2 3 0 0 0 1 1 0 2 S 2 P 2 6 1 0 0 0 0 0 1 S 2 P 6 2 1 0 1 0 0 0 2 F A 2 3 4 0 1 0 0 1 1 3 F A 2 3 5 1 1 0 1 0 0 3 F A 2 4 5 0 1 1 0 0 0 2 DefineCS2(hk) = S1P(1) & S1P(2) | S2P(hk) & S2P(kh), S 2 P 1 3 0 0 0 1 1 S 2 P 2 4 0 0 1 0 1 S 2 P 3 4 0 1 0 0 1 S 2 P 4 3 1 0 0 0 1 S 3 P 1 2 3 0 0 0 1 0 0 1 S 3 P 1 2 5 0 0 0 1 0 0 1 S 3 P 1 6 3 0 0 0 1 0 0 1 S 3 P 4 3 2 1 0 0 0 0 0 1 S 3 P 4 3 6 1 0 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 0 1 S 3 P 6 1 2 0 0 0 0 1 0 1 S 3 P 6 3 2 0 0 0 0 1 0 1 S 3 P 6 3 4 0 0 0 0 1 0 1 S 3 P 5 2 1 0 0 0 0 0 1 1 S 3 P 5 2 3 0 0 0 0 0 1 1 S 3 P 5 4 3 0 0 0 0 0 1 1 CS0 says all edges are equal. S 3 P 1 3 4 0 1 0 0 S 3 P 2 4 3 1 0 0 0 CS2: 16 45 are best, 12 25 34 36 are 2nd best, 23 worst. I like it 4cycle with 2 1hairs is best. 4cycle with 1 2hair 2nd best 6cycle worst

  36. ESP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E Analyst TickerSymbolmatrix w/0 labels (1 = “recommends”) 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 An 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 WSP2 3 3 6 4 8 8 a e c 5 4 6 3 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ESP2 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 9 9 9 9 9 e e e e 9 9 9 9 9 18 16 18 16 12 16 16 17 18 18 17 17 18 18 18 17 13 13 WomenSet ARM: MinSup=6 Mincnf=.75 EventSet ARM MnSp=9 Mncf=.75 Frequent 1WomenSets: 1 2 3 4 12 13 14 Frequency (#events attended) 8 7 8 8 6 7 8 Freq 1EventSets: 3 5 6 7 8 9 c Freq (# attended) 6 8 8 10 14 12 6 Candidate 2WomenSets: 12 13 14 1c 1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #events co-attended 6 7 7 2 2 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 Cand 2EventSets: 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 8c 9c Freq=#attended 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 5 5 Frequent 2WomenSets: 12 13 14 1c1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #events co-attended 6 7 7 22 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 freq 2EventSets: 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 Freq=#attended 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 Cand 3EventSets 568 578 all others excl because a sub2 not freq Freq # attended 6 5 Cand3WSets: 123 124 134 234 (cde is excluded since ce is infreq) Freq #events co-attended 5 5 6 5 Strong Erules 35 53 56 65 57 58 68 78 98 567 657 567 576 675 (Says 567 is a strong Event community?) Frequent 3WomenSets:123 124 134 234 Freq #events co-attended 5 5 6 5 Freq 3ESets: 567 Freq=6 5 StrongWrules21 12 13 31 14 41 23 32 24 42 34 43 134 314 413 134 143 341 Says 1234 is a strong Women community? Confidence: .83 .75 .87 .87 .87 .87 .83 .75 .83 .75 .87 .87 .75 .75 .75 .83 .83 .83 But 134 is a very strong Women Commun? Note: When I did this ARM analysis, I had several degrees miscounted. None-the-less, I think the same general negative result is expected. Next we try using the WSP2 and ESP2 relationships for ARM??

  37. A Kcliqueand a 3clique that shares an edge (and thus 2 vertices) form a (K+1)clique iff The K-2 edges between the non-shared 3clique vertex and each of the K-2 non-shared Kclique vertices exists. G7 The 1st time no 3clique shares an edge with a Kclique, the Kclique is maximal. Find a Maximal Maximal Clique for each v (a MaxClique containing v with max # of vertices) E 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3Cliques as a set of vertex triples 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 3 4 8 14182022 1 4 8 9 14 1 3 8 1314 1 1 1 1 1 1 1 1 1 1 1 1 1 5 5 6 6 7 7 8 8 8 9 1111 13 7 11 7 11 5 6 2 3 4 3 5 6 4 1 1 1 1 1 1 2 2 2 2 2 2 2 2 141414 18 20 22 3 3 3 3 4 4 4 4 2 3 4 2 2 2 1 4 8 14 1 3 8 14 2 2 2 2 2 2 2 2 2 3 3 3 3 8 8 8 141414 18 20 22 4 4 4 4 1 3 4 1 3 4 1 1 1 1 2 8 14 3 3 3 3 3 3 3 3 3 4 4 4 4 8 8 8 9 9 141414 33 8 8 8 13 1 2 4 1 33 1 2 4 9 1 2 3 1 4 4 4 5 6 6 7 9 9 9 9 9 141414 11 7 7 17 3131 3333 34 1 2 3 3 1 17 6 3334 3 31 31 24 2424 24 2424 25 25 26 27 28 3030 33 3434 26 32 32 30 34 3334 30 2830 32 26 25 34 27 28 2929 3030 3131 32 34 34 3234 3334 3334 34 30 24 3432 2428 9 9 29 18 20 22 14 4 8 2 1 3 12 30 31 10 33 19 21 15 16 27 34 17 23 26 24 29 28 11 32 25 5 7 6 9 13 16 9 10 6 3 4 4 4 5 2 3 1 2 52 2 2 2 2 32 22 53 3 2 43 4 4 6 11 16 Remaining pairwise ANDs after removal of PURE0 pairwise ANDs (i.e., after CS0). So these are the 3cliques in pTree form. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 111314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 28303334 2632 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

  38. Find Maximal Cliques 1. If a 3cliques shares nothing with any other 3clique, then it is maximal, else: G7 2. A 3cliques that shares an edge with a 3clique form a 4clique iff the 6th unknown edge exists. A Kcliqueand a 3clique that shares an edge (and thus 2 vertices) form a (K+1)clique iff The K-2 edges from the one non-shared 3clique vertex to the K-2 non-non-shared Kclique vertices exist. 3. A 4clique and a 3clique that shares an edge form a 5clique iff 9th,10th edges exist. 6. The 1st time no 3clique shares an edge w a Kclique, the Kclique is maximal. Remove participating 3cliques from the list and start over? E 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4. A 5clique and a 3clique that shares an edge form a 6clique iff the 13th 14th and 15th unknown edges exist. Unique 3Cliques (as sets) 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 8 14 18 20 22 4 8 9 14 1 1 1 1 1 1 1 2 2 2 4 4 4 5 5 6 6 3 3 3 8 13 14 7 11 7 11 4 8 14 2 2 3 3 3 6 9 9 4 4 4 4 9 7 31 31 6 14 8 14 33 17 33 34 24 24 24 25 27 29 28 30 30 26 30 32 34 33 34 32 34 34 … After finishing 4clique search, do 5clique with 1234 and 128 by check existence of 38 48, y y so 12348 5clique. 1 2 3 4 y 1 2 3 8 y 1 2 3 e y 1 2 4 8 y 1 2 4 e y 18 20 22 14 1 8 4 3 1 2 4 5 2 3 3 4 2 5 3 1 2 1 3 4 12 16 33 34 30 17 15 10 21 19 27 31 23 32 28 26 24 25 11 29 5 7 6 9 13 1234, 12e check 3e 4e, y y so 1234e 5clique … After finishing 5clique search, do 6clique 12348 and 12e by check existence of 3e 4e 8e, y y n, so 12348e not 6clique; 1234e and 128 by check existence of 3e 4e 8e, y y n, so 12348e not 6clique. And no other 6 cliques contains 12? 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 We haven’t used pTrees! One needs to study the literature on how maximal cliques are typically mined! Notes: One cannot remove a clique just because it is maximal? So what do we do once we discover 12348 and 1234e as maximal 5cliques? Do we have to retain all the 3 cliques and start over? Or? Would it suffice to find one maximal clique containing each vertex? Or find the maximal maximal clique (a maximal clique containing v that has the maximum number of vertices) containing each vertex? Remaining edges after CS0 (removal of PURE0 pairwise ANDs). So these are the 3cliques in pTree form. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 111314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 28303334 2632 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

  39. TS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ANalystTickerSymbolRelationship w/0 labels (1 = “recommends”) 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 AN 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 3 3 6 4 8 8 a e c 5 4 6 3 3 AnalystSet ARM: MinSup=6 Mincnf= .75 Large 1AnalystRecomendedSets: 1 2 3 4 c d e Frequency (# of stock recommended) 8 7 8 8 6 7 8 StockRecommendedSet ARM MnSp=6 Mncf=90% Frequent 1 StockRcommendedSets: 3 5 6 7 8 9 c Frequncy (# of Analyst Recommending) 6 8 8 10 14 12 6 Candidate 2AnalystSets: 12 13 14 1c 1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #stock recommendd 6 7 7 2 2 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 Cand 2StockSets 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 8c 9c Freq = # of AN: 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 5 5 Frequent 2AnalystSets 12 13 14 1c1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #Stock recomm 6 7 7 22 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 freq 2StockSets 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 Freq= # of AN: 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 Cand3AnalystSets: 123 124 134 234 (cde is excluded since ce is infreq) Freq #Stock recom 5 5 6 5 Candidate 3StockSets 568 578 (all others excluded due to a 2subset not freq) Frequency = # of AN: 6 5 Frequent 3AnalystSets:123 124 134 234 Frequency: #Stock Recommended 5 6 5 Frequent 3StockSets: 568 Frequency= # of AN= 6 5 Strongrules21 12 13 31 14 41 23 32 24 42 34 43 134 314 413 134 143 341 Analysts 1,3,4 seem to be most in synch, Conf .83 .75 .87 .87 .87 .87 .83 .75 .83 .75 .87 .87 .75 .75 .75 .83 .83 .83 Ruleconf%Supp(#ofAN)AntecedentSize 35 100 6 1 53 75 6 1 56 75 6 1 65 75 6 1 57 75 6 1 75 60 6 1 58 88 7 1 85 50 7 1 68 88 7 1 86 41 7 1 78 80 8 1 87 86 8 1 89 64 9 1 98 75 9 1 568 75 6 1 658 75 6 1 856 43 6 1 568 100 6 2 586 86 6 2 685 86 6 2 I think Antecedent Size is important. We can think of these as rule labels. My favorite rule is 568 since it has hi confidence And hi Antecedent Size (+ decent Support). We could rate Analysts and use weighted Counts as Frequency (vertex label We can use Sentiment as stock weights and build it into confidence (or use antecedent SA as a column

  40. AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS ANalystTickerSymbolRelationship w labels (1 = “recommends”) 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 B v w s s t t u q c m d a a q D 0 1 1 1 1 1 0 1 1 0 0 0 0 0 F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A 2 3 6 4 8 8 a e c 5 4 6 3 2 C 5 3 7 8 2 3 7 6 9 2 9 9 8 4 E 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 G 9 5 8 8 2 1 8 8 5 7 4 5 6 3 9 9 6 7 TS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 AN 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 3 3 6 4 8 8 a e c 5 4 6 3 3 (A,B,C,D) is a 4 attribute Vertex Label (on the Ticker Sysmbol part only). Any features of Ticker Symbols can be included here. A is the 1-counts of the Analysts pTrees (number recommending that stock in hexadecimal, These 1-counts are also given in the blue row in the other matrix). C is a decimal column which gives a 1-9 rating of stock performance during the previous week. B is a character column categorizing stocks by type. D is a binary column indicating whether the stock trades on the Nikkei (yes| no). Conditions on these label columns (e.g., expressed in SQL) give us a pTree mask to implement the condition. Likewise (E,F,G) is a 3 attribute Vertex Label (on Analyst part). Any Analyst (or Investor) feature can be included. E is the 1-counts of stock pTrees (# of Analysts recommending that stock (also list in the other matrix). G is decimal giving the Analysts yearly salary in billions. F is binary indicating whether the Analyst is male (yes| no). Conditions on these labels (e.g., expressed in SQL) give us a mask to apply to our pTrees to implement the condition. Probably the simplest implementation language for this recommender would be [PL] SQL or MySQL. We would have only two tables: Stock Table: with the first 18 columns being the AN pTrees and the final 4 columns being A B C and D. Analyst Table: with the first 14 columns being the TS pTreesand the final 3 columns being E F and G. We can call for ANDs, Ors, COMPLIMENTs etc, from SQL! Anyone can program SQL, right? Maybe R would be a good language so we could have one table that can be rotated???

  41. my ideas on what the book paper might contain that would push it beyond the workshop paper. Expansion Idea 1:  The reviewers asked for more charts and graphs - i.e., more performance studies comparing to the competition. Expansion Idea 2:  Expand to include a labelled bipartite graph (the disjoint vertex sets” "Ticker Symbol" and "Analyst" and an edge connecting, e.g., AAPL with Buffet iff a Buffet tweet was sentiment rich (positive or negative) regarding AAPL.  (So an undirected edge connects an Analyst with a Ticker Symbol iff there is a pertinent tweet from the Analyst on that stock on that day/week Analysts (AN) would be labelled (vertex label) by "Respect Level". Stock Ticker Symbols (TS) would be labelled (vertex label) by "buy-sell" value (a positive number if buy and a negative number if sell) as recommended by various stock rating entities (e.g., There are known raters that label stocks as Strong Buy, Buy, Hold, Sell, Strong Sell which would be 2,1,0,-1,-2 respectively) Tweets are labelled (edge label) by Sentiment (LoValue for very negative sentiment and HiValue for very positive sentiment), as already produced by various software products (e.g., MS Azure S?) This set up (vertex labelled and edge labelled bipartite graphs) would allow us to try lots of pTree tools: ARM using pTrees; just like MBR except instead of customers and products we have Analysts and Ticker Symbols...) , Clustering using pTrees (what we are currently looking at at our Saturday meetings), Outlier Analysis using pTrees (also related to current Saturday topics)... Community Detection using pTrees (related to clustering and Outlier Analysis) From Arijit Sept 25, 2015 I am leveraging Microsoft's Azure Machine Learning Service for the Sentiment Analysis and have worked to build a better sentiment analyzer on Tweet Data but the Service end point is public and can be used for research. So far: • I have the pulled tweets from Twitter of all the investors whom we would like to track over last 5 years. I have coded around the Twitter platform limitations and the code is parallelized with multiple configurations. This data stored on an Azure SQL Database. • I have the Azure Sentiment Analysis Service running on this pulled tweets and have the sentiment score from every tweet of all these individuals.  ( This was not done in the workshop paper) • I have a separate service which runs which queries the Yahoo Finance API and has pulled the historical data for every ticker symbol. This data is also stored in a separate table. • I have added the average sentiment score (SScore) for a particular day for a particular ticker symbol and have mapped this information to the other fundamentals of the stock - like Open Price, Closed Price, Volume, P/E ratio. ( This was also not done in the workshop paper) • With this data we can plugin various algs and test My main Hypothesis in naïve form is: Social Sentiments have an effect on stock fluctuations. We can do all sorts of variations and then test different algorithms. One approach I have been thinking is how to weigh the investors, rather than just having an Average Sentiment Score have a Weighted Average Sentiment Score. • I also have the code for the Exponential Moving Sentiment Score ( EMScore) calculated in various time periods like 15day Moving Average, 200 day MA, 1 year and so on so that I can see which of these measures is the best indicator of the volatility of the tickers the most.  I have attached a spreadsheet which gives an idea on what data is been captured now. This is just a sample sheet which I was working to test the EMScore on Tickers NFLX and AAPL between 07/01 till 07/30. On Sheet3 you would see a column with the DailyAvgSentimentScore values included. The Azure Sentiment Analyzer service consumes the data on Sheet 4 for every tweet from every user and generates a sentiment score. The average of all these Sentiment Scores are calculated and the Daily Avg Sentiment Score field is populated on Sheet 3. Sheet 2 shows the EMScore calculation and you can see a chart which shows how the Sentiment Score is varying over the period of time and how its sort of following the same trend as the Ticker Closing prices.

  42. MCS + DND2 (MCS= Minimum Common Cousins) DND is all but 1 2 3 4 8 14 33 34 1 2 3 4 8 143334 1 1 1 1 1 2 2 2 2 3 3 3 4 4 2 3 4 8 14 3 4 8 14 4 8 14 8 14 5 5 5 5 4 4 0 0 4 4 4 3 3 4 4 3 3 4 3 3 3 3 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 3 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 4 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 8 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 14 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DND all but 1 2 3 4 33 34 1 2 3 4 3334 1 1 1 2 2 3 2 3 4 3 4 4 3 3 3 3 0 0 2 2 2 2 2 2 1 0 1 1 1 0 0 0 0 0 1 1 1 2 1 0 1 1 0 0 0 1 1 0 0 1 3 1 1 0 1 0 0 1 0 1 0 1 0 4 1 1 1 0 0 0 1 1 0 1 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 DND is all but-> 1 2 3 4 8 9 14 33 34 1 2 3 4 8 9 143334 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 9 2 3 4 8 14 3 4 8 14 4 8 9 14 8 14 33 5 5 6 5 4 2 4 1 0 4 4 4 3 3 4 4 3 3 4 3 0 3 3 3 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 2 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 3 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 4 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 8 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 9 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 33 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 That also deletes (1 6) (1 7) (and may delete others as well). This is a good clustering! 1234 is a 4clique! Nothing will change after this round. Notes: The only way (3 10), (29 34) will get deleted is with a final round that deletes any edge whose endpoints satisfy CS=CC=0 (+DNI). 1 2 3 4 5 6 7 8 9 11142430313334 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 5 5 6 6 9 9 9 242424 3030 3131 DND1012131516171819202122232526272829322 3 4 5 6 7 8 9 1114 3 4 8 14 4 8 9 1433 8 14 7 11 7 11 313334 303334 3334 3334 105 7 5 3 3 3 4 5 3 4 3 3 3 5 4 4 5 4 2 2 2 3 1 2 3 4 4 3 3 4 3 2 3 1 3 3 1 1 1 1 2 2 1 2 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 30 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 33 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 24 31 30 6 7 11 5 1 2 3 4 5 6 7 8 9 11142024252628293031323334 DND 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 5 5 6 6 9 9 9 242424 3030 3131 32 2 3 4 5 6 7 8 9 1114 3 4 8 14 4 8 9 1433 8 14 7 11 7 11 313334 303334 3334 3334 34 10 Sum125 7 5 3 3 3 4 5 3 4 2 3 1 1 2 2 3 3 1 5 5 4 5 4 2 2 2 3 1 2 3 4 4 3 3 4 3 2 3 1 3 3 1 1 1 1 2 2 1 2 1 1 1 1 1 1 0 12 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 13 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 15 3 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 16 4 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 17 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 8 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 21 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 14 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 25 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 26 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 27 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 33 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 28 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 0 29 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 DONOT DELETE-2:: As a first step, create (kept current) a DoNotDelete (DND) list. Include all vertices with a 1 or a 2 count. Keep DND current by adding vertices as soon as they exhibit the “Delete” condition. Applying DND first means we only AND edge endpoints that are both off the DND (reducing the AND burden considerably). 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 5 5 6 6 9 9 9 14 20 2424242424 252525 26 28 2929 3030 3131 3232 2 3 4 5 6 7 8 9 11142032 3 4 8 142031 4 8 9 14282933 8 14 7 11 7 11 313334 34 34 2628303334 262832 32 34 3234 3334 3334 3334 169 106 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 1116 7 5 5 2 2 2 3 1 2 3 1 0 4 4 3 3 1 0 4 3 2 3 0 0 1 3 3 1 1 2 1 2 2 1 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 2 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 D 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 N 2 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 D 3 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 8 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 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