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

4. What Vertices are Most Important?
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 Ordinary Description Possible Network Interpretation
prominent
Vertex is “visible” to many other vertices
admired
Vertex is “chosen” by many other vertices
listened to
Vertex is “received” by many other vertices
in the know
Vertex is short distance from many sources of information
linchpin
Vertex irreplaceable part
gate keeper
Vertex stands between one part of graph and another
involved
Vertex connected to many parts of graph

7. Davis Southern Women Degree Centrality

8. Davis’ Southern Women - Centrality

9. A Simple Network A B C D E F G - 1

10. 𝐶 𝐷 𝑣 𝑖 = 𝑘 𝑖
centrality
degree

11. Degree Centrality can Fail to DifferentiateB C D E F G - 1 CD A 4 B C D E F G 1

12. Degree Centrality Can Mislead

13. 𝐶 𝑐 𝑣 𝑖 = 1 𝑗=1 𝑛 𝑑( 𝑣 𝑖 𝑣, 𝑣 𝑗 )
centrality
closeness

14. Closeness Centrality Closeness = 1/total distance to other vertices
𝐶 𝑐 𝑣 = 1 𝑖=1 𝑛 𝑑(𝑣, 𝑣 𝑖 )
𝐶 𝑐 𝐴 = 1 𝑑 𝐴𝐵 +𝑑 𝐴𝐶 +𝑑 𝐴𝐷 +𝑑 𝐴𝐸 +𝑑 𝐴𝐹 +𝑑 𝐴𝐺
𝐶 𝑐 𝐴 =
𝐶 𝑐 𝐴 = 1 8 =0.125

15. Compare Two Graphs What is the problem here? How would you fix it?
Compute Closeness
Centrality of a Vertex
𝐶 𝑐 𝐴 = =0.3 3
𝐶 𝑐 𝐴 = =0.2
What is the problem here?
How would you fix it?

16. Normalization 𝐶 ′ 𝐶 𝑣 𝑖 = 𝑛−1 𝑗=1 𝑛 𝑑 𝑣 𝑖 , 𝑣 𝑗
Adjusting a formula to take into account things like graph size
Usually by “mapping” values to (0…1) or -1…+1
For closeness centrality:
𝐶 ′ 𝐶 𝑣 𝑖 = 𝑛−1 𝑗=1 𝑛 𝑑 𝑣 𝑖 , 𝑣 𝑗
Where n is number of vertices in the graph

17. Compare Two Graphs
𝐶′ 𝑐 𝐴 = =1
𝐶′ 𝑐 𝐴 = =1
Intuitively, both blue vertices should have the same closeness
centrality since both are 1 step away from all other vertices.

18. 𝐶 𝑏 𝑣 𝑖 = 𝑛 𝑠𝑡𝑖 𝑔 𝑠𝑡
centrality
Betweenness

19. Betweenness Centrality
Fraction of shortest paths that include vertex A B C D E F G -
1,1
2,4
2,1
3,4

20. Betweenness Centrality
Fraction of shortest paths that include vertex
1 shortest path of 4 goes through A
𝐶 𝑏 𝑣 𝑖 = 𝑛 𝑠𝑡𝑖 𝑔 𝑠𝑡 A B C D E F G -
1,1
2,4
2,1
3,4
Example: Calculate betweenness centrality of vertex A
1 shortest path of 4 goes through A
𝐶 𝑏 𝑣 𝑖 = 𝑛 𝑠𝑡𝑖 𝑔 𝑠𝑡 = = 0.75
1 shortest path of 4 goes through A

21. 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
𝐶′ 𝐵 = 𝐶 𝐵 𝑛−1 𝑛−2 2

22. Calculate Cb(F) A B C D E F G - 1 4

23. Vertex Centrality Comparison
Usually centrality metrics positively correlated
When not, something interesting going on
Low Degree
Low Closeness
Low Betweenness
High
Degree
Ego embedded in cluster that is far from the rest of the network
Ego's connections are redundant - communication bypasses him/her
High Closeness
Ego tied to important or active alters
Probably multiple paths in the network, ego is near many people, but so are many others
High Betweenness
Ego's few ties are crucial for network flow
Very rare cell. Would mean that ego monopolizes the ties from a small number of people to many others.

24. 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)

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

26. Diagrams by J Moody, Duke U.

27. centrality
Eigenvector

28. Consider this Example
The two red nodes have similar amounts of “local” centrality, but different amounts of “global” centrality.

29. 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) )

30. 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

31. 1

32. 2 3 5 4

33. 6 7 13 10 18 9

34. 15 22 33 16 20 52 36 46 25

35. 40 58
126 52 72
139 94
209 92

36. And consider the matrix What does this matrix “do” to the vector ?
Consider the xy coordinate plane where a line from (0,0) to (x,y) is the vector
And consider the matrix
What does this matrix “do” to the vector ?
(x,y) x y y x 1 1

37. Matrix Multiplication as Distortion1 1 1
BUT 1 1 1

38. So, what is an Eigenvector?

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

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

41. 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.

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

43. 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.

44. Intuitiveness Visible on Rearrangment

45. Eigenvector vs. Power (Bonacich)

46. Centrality in Social Networks
Power / Eigenvalue
In recent work, Borgatti (2003; 2005) discusses centrality in terms of two key dimensions:
Substantively, the key question for centrality is knowing what is flowing through the network. The key features are:
Whether the actor retains the good to pass to others (Information, Diseases) or whether they pass the good and then loose it (physical objects)
Whether the key factor for spread is distance (disease with low pij) or multiple sources (information)
The off-the-shelf measures do not always match the social process of interest, so researchers need to be mindful of this.

47. What Can We Study with Centrality?
City systems
Illegal networks
Marketing targets
Opinion formation/spread
Epidemiology

48. Directed Networks: Centrality v. Prestige
An actor is considered central if her ties make her visible to others
Visibility by direct ties AND by indirect ties through intermediaries
Many social and economic phenomena such as access and control over resources and brokerage of information involve centrality since simply participating in interaction is what counts.
But the number of ties alone does not determine importance
So we distinguish a second type of visibility : prestige
Takes into account the direction of the tie
Generally, more incoming ties  more prestige
And, more incoming ties from higher prestige vertices  more prestige
Based on Aliseya Wright HCC Spring 2005 Wednesday, April 13,

49. Google(PageRank): Overview
Pre-computes a rank-vector
Provides a-priori (offline) importance estimates for all pages on Web
Independent of search query
In-degree  prestige
Not all votes are worth the same
Prestige of a page is the sum of prestige of citing pages: p = Ep
Pre-compute query independent prestige score
Query time: prestige scores used in conjunction with query-specific IR scores
Mining the Web
Chakrabarti and Ramakrishnan

50. Chakrabarti and Ramakrishnan
Google(PageRank)
Assumption
the prestige of a page is proportional to the sum of the prestige scores of pages linking to it
Random surfer on strongly connected web graph
E is adjacency matrix of the Web
No parallel edges
matrix L derived from E by normalizing all row-sums to one: .
Mining the Web
Chakrabarti and Ramakrishnan

51. Chakrabarti and Ramakrishnan
The PageRank
After ith step:
Convergence to
stationary distribution of L.
p -> principal eigenvector of LT
Called the PageRank
Convergence criteria
L is irreducible
there is a directed path from every node to every other node
L is aperiodic
for all u & v, there are paths with all possible number of links on them, except for a finite set of path lengths
Mining the Web
Chakrabarti and Ramakrishnan