Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale

1. Network Theory:Computational Phenomena and Processes • Social Network Analysis Dr. Henry HexmoorDepartment of Computer ScienceSouthern Illinois University Carbondale

2. Degree, Indegree, Outdegree Centrality Degree Centrality: Indegree Centrality: OutdegreeCentrality:

3. Eigenvector Centrality=CE (i)=I’th entry of eigenvector e e = largest eigenvalue of adjacency matrix

4. BetweennessCentrality Normalized betweenness= = number of geodesic linking across i and j has pass through node k.

5. Closeness Centrality Scaling factor Adjustment

6. One-node network 1-Connection 0-No connection One-mode network: Actors are tied to one another considering one type of relationship; i.e Binary Adjacency matrix

7. Two-node network Two-node network: Actors are tied to events. • Incident network • Bipartite graph e.g. Student attending classes

8. Affiliation network Actors are tied to ----Organization/Attributes; e.g. Affinity network, Homophily network Sociogram ≡ Org1……………………..…….Org n Attribute m1………………..Attribute mn 12..n 12..n Attributes People { } Points-------------individuals Lines --------------relationship

9. Centrality Star graph A has higher degrees. A is central to all. Centrality: Quantifying a network node. (i)= ij

10. E A C Normalized Centrality Centrality:Normalized Centrality: A is more central than F B F D (A) ‘ 80% = = (A)= (A)=4 6-1 (D)=2 (B)=3 (C)=2 (E)=2 (F)=1 , , , ,

11. E A C Directed network centrality: Prestigue of A=B=C=E=2 (Indegree) F D B (A)=3 (A)=2 (B)=2 (F)=1 (F)= (D)=2 (E)=2 (A) (Normalize centrality) ‘ (A)= = = 40% 6-1

12. Eigen vectors Vector X is a matrix with n rows and column, linear operator A, maps the vector X to matrix product AX A

13. Eigen Value to the equations

14. Second degree centrality I Consider this 16 degree graph network: H B G M A D C N O E L K F J Eigen value centrality (A)= (O)=6 higher Eigen values

15. Betweenness Centrality Betweennes centrality measures the extent to which a vertex lies on paths between other vertices. =Number of paths from i to j passing through k = Number of shortest distance path from i to j K= Geodesic distance (K)=

16. E A C Normalized centrality: F B D =

17. Closeness Centrality Closeness centrality is the mean distance from a vertex to other vertices. E A C F B D (i)= (i)= f= farness c= closeness d= distance between i & j n= total number of nodes

18. Eigen vector Eigen vector for N(i) = Neighbours of i = {J } where N=(, ) Eigen vector centrality: Therefore, (i)= .

19. Page Rank Cetrality The numerical weight that it assigns to any given element E is referred to as the PageRank of E and denoted by PR(E). Page Rank Centrality: (i)=

20. Bonacich/Beta Centrality • Both centrality and power were a function of the connections of the actors in one's neighborhood. • The more connections the actors in your neighborhood have the more central you are. • The fewer the connections the actors in your neighborhood, the more powerful you are. • It is the weighted centrality

21. Bonacich/Beta Centrality: = Here, (local importance)

22. Density Density: It is the level of ties/connectedness in a network; It is a measure of a network’s distance from a complete graph. Complete graph: Every node is connected to every node in the network

23. Ego Density L = number of links in network n = number of nodes in the network

24. Structural Hole (Ron Burt) Let’s consider this, • The gap between connected components is the hole • Structural hole provides diversity of information for nodes that bridge them • Without structural hole information becomes redundant and less available 1 gap 2 Structural Hole

25. Brokering Brokering is bridging different group of individuals. 1.Coordinator (local brokers; Intragroup brokering) e.g. manger, mediating employees 2. Consultant (Intergroup brokering by an outsider e.g. middle man in business between buyers &seller, stock agent ) C B Bas Coordinator/ Broker A Buyer Consultant Seller

26. Brokering 3. Representation (represents A when negotiating with C) e.g. hiring a mechanic to buy car for you 4. Gate Keeper(e.g a butler, chief of staff) A C B Actor Producer Agent Producer Actor

27. Dyadic Relation A B Dyads: Triads: when a triad consists of many ties, an open triad (triangle) is forbidden. A B B A A B A B 0 C

28. Triad Relations (census)

29. Components Component is a group where all individuals are connected to one another by at least one path. • Weak Component: A component ingoing direction of ties. • Strong Component: A component with directional ties. • Clique: A subgroup with mutual ties of three or more. who are directly connected to one another by mutual ties

30. Bonacich Centrality CBC = Degree Centrality • High Degree + Low Betweenness : Ego Connection are redundant • Low Closeness+ High Betweenness : Rare node but pivotal to many • In triads, there is a structural force toward transitivity.

31. Bonacich Reverse distance: Principle ofstrength of weak tie.(Granovetter, 1973): There is a social force that suggests transitivity. If A has ties to B and B to C, then there is tie from A to C.

32. Integration: Reverse distance Network distance Degree to which a node’s inward ties integrate it into the network.

33. Radiality Degree to which a node’s outward ties connects the node with novel nodes.

34. Edge between-ness Number of shortest path from s to t that pass through edge e Number of shortest path from s to t • This is important in diffusion studies like epidemics

35. Social Capital The network closure argument: Social Capital is created by a strongly interconnected network. The structural hole argument: Social Capital is created by a network of nodes who broker connections among disparate group.

36. Structural Equivalence= similarity of position in a network Euclidean Distance E.g., B E A C D have distance zero