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Centrality

Centrality. Spring 2012. Why do we care?. Diffusion (practices, information, disease) Structure, status, prestige Seeing, perspective, worldview Power as relational, constraints as relational Network location as dependent variable Explaining outcomes Supporting strategic “networking”.

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Centrality

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  1. Centrality Spring 2012

  2. Why do we care? • Diffusion (practices, information, disease) • Structure, status, prestige • Seeing, perspective, worldview • Power as relational, constraints as relational • Network location as dependent variable • Explaining outcomes • Supporting strategic “networking”

  3. Example: 2-Step Flow of Communication* • Micro- macro- link in communications theory • Lazarsfeld on mass media and voting (1940s) • high centrality nodes – opinion leaders – mediate broadcast info flow • later (Lazarsfeld & Katz (1955)) formalized as two-step flow of communication model: mass media messages filtered through more-exposed central members of social groups. *Remix of http://www.soc.umn.edu/~knoke/pages/SOC8412.htm

  4. The Question What Vertices are Most Important?

  5. Everyday Understandings • Important = prominent • Important = admired • Important = linchpin • Important = listened to • Important = in the know • Important = gate keeper • Important = involved

  6. Translations

  7. A Simple Network

  8. centrality degree

  9. Degree Centrality can Fail to Differentiate

  10. Degree Centrality Can Mislead

  11. centrality closeness

  12. Closeness Centrality • Closeness = 1/total distance to other vertices

  13. Compare Two Graphs • What is the problem here? • How would you fix it? Compute Closeness Centrality of a Vertex

  14. Normalization • Adjusting a formula to take into account things like graph size • Usually by “mapping” values to (0…1) or -1…+1 • For closeness centrality: • Where n is number of vertices in the graph

  15. Compare Two Graphs Intuitively, both blue vertices should have the same closeness centrality since both are 1 step away from all other vertices.

  16. centrality Betweenness

  17. Betweenness Centrality • Fraction of shortest paths that include vertex

  18. Betweenness Centrality • Fraction of shortest paths that include vertex 1 shortest path of 4 goes through A Example: Calculate betweenness centrality of vertex A 1 shortest path of 4 goes through A 1 shortest path of 4 goes through A = 0.75

  19. Normalizing Betweenness • Middle vertices should have same CB? • Since number of paths vertex COULD be on is (n-1)(n-2)/2 we can use this as our denominator

  20. Calculate Cb(F)

  21. Vertex Centrality Comparison • Usually centrality metrics positively correlated • When not, something interesting going on

  22. Information Centrality • Betweenness only uses geodesic paths • Information can also flow on longer paths • Sometimes we hear it through the grapevine • While betweenness focuses just on the geodesic, information centrality focuses on how information might flow through many different paths, weighted by strength of tie and distance. (Moody)

  23. Information Centrality Chapter 2 Resistance Distance, Information Centrality, Node Vulnerability and Vibrations in Complex Networks by Ernesto Estrada and Naomichi Hatano

  24. Diagrams by J Moody, Duke U.

  25. centrality Eigenvector

  26. Consider this Example • The two red nodes have similaramounts of “local” centrality,but different amounts of “global”centrality.

  27. Power/Eigenvector Centrality • Weakness of degree centrality – it counts your neighbors but not whether or not they count • Basic idea ego’s centrality is function of neighbors’ centrality C(ego) = f (C(ego’s neighbors) )

  28. Algorithm • Assume all vertices have centrality, C = 1 • Recalculate C by summing C of neighbors • Repeat the process • Each time we are “taking into account” the centralities of yet another “layer” of the vertices around us

  29. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  30. 2 2 2 2 2 2 3 5 4 2 3 4 4 2 2 2 4 5 2 2 2 2 2

  31. 7 7 7 6 6 6 7 13 10 6 13 9 18 6 6 6 10 13 6 6 7 7 7

  32. 20 20 20 16 16 15 22 52 36 16 33 25 46 16 15 15 36 52 16 16 20 20 20

  33. 72 72 72 52 52 40 58 139 94 52 126 92 209 52 40 40 94 139 52 52 72 72 72

  34. Consider the xy coordinate plane where aline from (0,0) to (x,y) is the vector • And consider the matrix • What does this matrix “do” to the vector ? (x,y) y x

  35. Matrix Multiplication as Distortion BUT

  36. So, what is an Eigenvector?

  37. Eigenvector • Adjacency matrix redistributes vertex contents • Some vector of contents is in equilibrium • These are the eigenvector centralities

  38. What is an Eigenvector? • Consider a graph & its 5x5 adjacency matrix, A

  39. And then consider a vector, x… • a 5x1 vector of values, one for each vertex in the graph. In this case, we've used the degree centrality of each vertex.

  40. What happens when… • …we multiply the vector x by the matrix A? • The result, of course, is another 5x1 vector.

  41. Axx diffuses the vertex values • Look at first element of resulting vector • The 1s in the A matrix "pick up" values of each vertex to which the first vertex is connected • Result value is sum of values of these vertices.

  42. Intuitiveness Visible on Rearrangment

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