Fast algorithm for detecting community structure in networks.

1. Fast algorithm for detecting community structure in networks. M. E. J. Newman, (2004). Presented by Muad Abu-Ata

2. Community structure • groups of vertices within which connections are dense but between which they are sparser. • Within-group( intra-group) edges. • High density • Between-group( inter-group) edges. • Low density.

3. Community Structure

4. Real Word Networks • Internet • World Wide Web. • Citation Networks. • Transportation Network. • Email Networks. • Food Webs. • Social Networks. • Biochemical Networks.

5. Examples of Community Structures • Communities of biochemical network correspond to functional units of some kind. • Communities of a web graph correspond to sets of web sites dealing with a related topics.

6. Finding Community Structures • Divide the network into non-empty groups( communities) in such a way that every vertex belongs to one of the communities. • Many possible divisions could be done. • We need a good division. • Measurement of good division.

7. Community Detection Approaches • Graph partitioning approaches: • Spectral bisection • The Kernighan-Lin (KL) algorithm • Hierarchical clustering. • The algorithm of Girvan and Newman. • The Newman fast algorithm.

8. 3 1 4 2 5 is always eigenvector with eigenvalue 0. Spectral bisection • Eigen-vectors of the graph Laplacian. • L = D-A • A is the adjacency matrix • D is a diagonal Matrix of vertex degrees

9. 3 1 4 2 5 Bisect ! The eigenvector corresponding to the lowest eigenvalue must have both positive and negative elements. +ve: reasonably fast; O(n3) sparse matrix case, Lancozos method reduces it to approximately to

10. Spectral Bisection (Cont.) • Disadvantages: • It only bisects graphs into 2 communities. Division into a larger number of communities is usually achieved by repeated bisection, but this does not always give satisfactory results. • we do not in general know ahead of time how many communities we want to divide the graph into.

11. The Kernighan-Lin( KL) algorithm • Benefit function Q: the number of edges that lie within the two groups minus the number that lie between them. • user specify the size of the two groups A & B. • divide the vertices into the two groups randomly. • Calculate the ∆Q for all possible exchange pair from A and B. • Swap the pair that maximizes the change of Q. (greedy algorithm) • Repeat 3 & 4 until all vertices have been swapped once. (any vertex that has been swapped is never swapped. ) • Go back over the sequence of swaps and find the highest Q.

12. KL algorithm (cont.) Time complexity: O(n2). -ve: requires a priori what the size of the groups will be. Running the algorithm for all possible group sizes O(n3). The best values of Q are always achieved for very asymmetric trivial division.

13. Hierarchical clustering • develop a similarity (or dissimilarity) measure xijbetween pairs (i,j) of vertices. • Apply the hierarchical clustering and build the dendogram or tree. • Cross section the dendogram at any level will give the communities at that level.

14. Hierarchical clustering

15. Hierarchical clustering • Time complexity:O(n2logn) • N2 vertex pairs. • Calculations of all similarity measures takes َ O (mn). • Sorting N2 similarity measures takes O(n2logn) for sorting. • Constructing the dendogram takes linear time. • it doesn't require us to specify the size or number of groups we want to look for beforehand. • -ve: • It does not tell us how many groups should be used to get the best division of the network (Where to cut!).

16. A B Girvan and Newman( GN) Algorithm • Edge Betweeness: The number of shortest paths between vertex pairs that goes along an edge. • Calculate the betweenness for all edges in the network. • Remove the edge with the highest betweenness. • Recalculate betweennesses for all edges affected by the removal. • Repeat from step 2 until no edges remain. • cross cut the dendogram of components. • By removing these edges, we separate groups from one another as components.

17. The GN Algorithm

18. The GN Algorithm • Time complexity: • O(m2n) O(n3) • O( mn) for calculating edge betweeness. • m iterations. • -ve: • It provides no guide to how many communities a network should be split into (where to cross cut!). modularity measure.

19. Newman Fast Algorithm • Modularity Measure • the fraction of within-community edges minus the expected value of the same quantity for randomized network( edges fall at random with no regard to community structure) • Q= 0  no community structure. • 0.3<Q<0.7 significant community structure. • Generally the number of ways to divide n vertices into g non-empty groups is given by the Sterling number of the second kind S(n,g).The number of distinct community divisions is • Greedy approach to maximize Q.

20. Newman Fast Algorithm • Separate each vertex solely into n community. • Calculate ∆Q for all possible community pairs. • Merge the pair of the largest increase in Q. • Repeat 2 & 3 until all communities merged in one community. • Cross cut the dendogram where Q is maximum Notes: ∆Q=eij+ eji – 2aiaj Calculate ∆Q only for pairs that are connected by an edge.

21. Newman Fast Algorithm

22. Newman Fast Algorithm

23. Newman Fast Algorithm • Time Complexity • O((m+n)n) O(n2) for sparse graphs

24. Conclusion • Newman fast algorithm is: • considerably fast O(n2) • gives good divisions. • No need a prior knowledge of the community sizes. • No need a prior knowledge of the number of communities.

25. References • Fast algorithm for detecting community structure in networks, M. E. J. Newman. • Detecting community structure in network, M. E. J. Newman. • Finding community structure in very large networks, Aaron Clauset, M. E. J. Newman, and Cristopher Moore.