1. Centrality in Social Networks
Kristina Lerman
University of Southern California
Access to data allows us to ask new questions, empirically measure effects
CS 599: Social Media Analysis
University of Southern California

2. Network analysis

3. 4.2

4. Three views of social network analysis
Freeman, L “Centrality in Social Networks: Conceptual Clarification”, Social Networks 1, No. 3.
Bonacich, P “Power and Centrality in Social Networks: a Family of Measures”, American Journal of Sociology
Franceschetti, M “PageRank: standing on the shoulders of giants” Commun. ACM, Vol. 54, pp

5. Power in social networks
Certain positions within the network give nodes more power
Directly affect/influence others
Control the flow of information
Avoid control of others

6. Centrality in social networks
Centrality encodes the relationship between structure and power in groups
Certain positions within the network give nodes more power or importance
How do we measure importance?
Who can directly affect/influence others?
Highest degree nodes are “in the thick of it”
Who controls information flow?
Nodes that fall on shortest paths between others can disrupt the flow of information between them
Who can quickly inform most others?
Nodes who are close to other nodes can quickly get information to them

7. Degree centrality The number of others a node is connected to
Node with high degree has high potential communication activity 1 2 3 4 5
node
In-degree
Out-degree
Total degree 1 2 3 5 4 1 2 3 4 5

8. Mathematical representation and operations on graphs
Adjacency matrix A 1 2 3 4 5 1 2 3 4 5 1 1 2 3 4 5
node
In-degree
Out-degree
Total degree 1 2 3 5 4
Out-degree: row sum
In-degree: column sum

9. Betweenness centrality
Number of shortest paths (geodesics) connecting all pairs of other nodes that pass through a given node
Node with highest betweenness can potentially control or distort communication 1 2 3 4 5
12
123
124
1245
23
24
245
32
34
35 1 2 3 4 5
452
4523
45
52
523
524

10. Closeness centrality
Node that is closest to other nodes can reach other nodes in shortest amount of time
Can best avoid being controlled by others
Closeness centrality is sum of geodesic distances from a node to all other nodes

11. Self-consistent measures of centrality
Katz (1953): Katz score
“not only on how many others a person is connected to, but who he is connected to”
One’s status is determined by the status of the people s/he is connected to
Bonacich (1972): Eigenvector centrality
Node’s centrality is the sum of the centralities of its connections
e is the eigenvector of A, and l its associated eigenvalue (largest eigenvalue)

12. Alpha-centrality (Bonacich, 1987)
Similar to eigenvector centrality, but the degree to which a node centrality contributes to the centralities of other nodes depends on parameter a.
Mathematical interpretation
ci(a) is the expected number of paths activated directly or indirectly by a node i

13. A closer look at Alpha-Centrality1 2 3 4 5
Alpha-Centrality matrix
1st term: number of paths of length 1 (edges) between i and j
Contribution of this term to ci(a) is SjAij 1

14. A closer look at Alpha-Centrality1 2 3 4 5
Alpha-Centrality matrix
2nd term: number of paths of length 2 between i and j 1 1 1 2 x =

15. A closer look at Alpha-Centrality1 2 3 4 5
Alpha-Centrality matrix
3rd term: number of paths of length 3 between i and j 1 2 1 1 2 x =

16. A closer look at Alpha-Centrality
Alpha-Centrality matrix
Number of paths of diverges as length of the path k grows
To keep the infinite sum finite, a < 1/l1, where l1 is the largest eigenvalue of A (also called radius of centrality)
Interpretation: Node’s centrality is the sum of paths of any length connecting it to other nodes, exponentially attenuated by length of the path, so that longer paths contribute less than shorter paths

17. Radius of centrality
Parameter a sets the length scale of communication or interactions.
For a = 0, only local interactions (with neighbors) are considered
Only local structure is important
centrality is same as degree centrality

18. Radius of centrality
Parameter a sets the length scale of communication or interactions.
As a grows, the length of interaction grows
Global structure becomes more important
Centrality depends on node’s position within a larger structure, e.g., a community

19. Radius of centrality
Parameter a sets the length scale of communication or interactions.
As a  1/l1, length of interactions becomes infinite
Global structure is important
Centrality is same as eigenvector centrality

20. Normalized Alpha-Centrality [Ghosh & Lerman 2011]
Alpha-Centrality diverges for a > 1/l1
Solution: Normalized Alpha-Centrality
Holds for

21. Multi-scale analysis with Alpha-Centrality
Parameter a allows for multi-scale analysis of networks
Differentiate between local and global structures
Study how rankings change with a
Leaders: high influence on group members
Nodes with high centrality for small values of a
Bridges: mediate communication between groups
Nodes with low centrality for small values of a
But high centrality for large values of a
Peripherals: poorly connected to everyone
Nodes with low centrality for any value of a

22. Karate club network [Zachary, 1977]
instructor
administrator

23. Ranking karate club members
Centrality scores of nodes vs. a
Betweenness centrality of 17 is zero, as expected, but 25 and 26 have scores
similar to 31. 23

24. Florentine families in 15th century Italy

25. Ranking Florentine families

26. Summary
Network position confers advantages or disadvantages to a node, but how you measure it depends on what you mean by advantage
Ability to directly reach many nodes  degree centrality
Ability to control information  betweenness centrality
Ability to avoid control  closeness centrality
Self-consistent definitions of centrality
Node’s centrality depends on centrality of those it is connected to, directly or indirectly, but contribution of distant nodes is attenuated by how far they are
Attenuation parameter sets the length scale of interactions
Can probe structure at different scales by varying this parameter

27. PageRank: Standing on the Shoulders of Giants [Franceschet]
Key insights
Analyzes the structure of the web of hyperlinks to determine importance score of web pages
A web page is important if it is pointed to by other important pages
An algorithm with deep mathematical roots
Random walks
Social network theory

28. PageRank and the Random Surfer
Starts at arbitrary page I H F B C E L M D A G

29. PageRank and the Random Surfer
Starts at arbitrary page
Bounces from page to page by following links randomly I H F B C E L M D A G

30. PageRank and the Random Surfer
Starts at arbitrary page
Bounces from page to page by following links randomly
PageRank score of a web page is the relative number of time it is visited by the Random Surfer I H F B C E L M D A G

31. Mathematics of PageRank
PageRank is a solution to a random walk on a graph
Adjacency matrix of the graph A I H L M G E F B C D A A B C D E F G H I L M 1

32. Mathematics of PageRank
PageRank is a solution to a random walk on a graph I H L M G E F B C D A
(Diagonal) Out-degree matrix D A B C D E F G H I L M 1 2 3

33. Mathematics of PageRank
PageRank is a solution to a random walk on a graph
hij is probability to go from node i to node j
hij=1/di  H=D-1A I H L M G E F B C D A A B C D E F G H I L M 1 .5 .3

34. Mathematics of PageRank
PageRank of page j is defined recursively as
pj=Sipihij
Or in matrix form p=pH
What contributes to PageRank score?
Number of links page j receives
Cf B and D
Number of outgoing links of linking pages
Cf E’s effect on F and B’s effect on C
PageRank scores of linking pages
Cf E and B I H L M G E F B C D A

35. … but there are problems
Random Surfer gets trapped by dangling nodes! (no outlinks)
Solution: matrix S
replace zero rows in H with u=[0.9,0.9, …, 0.9]
From dangling node, surfer jumps to any other node
.09 1 .5 .3 I H L M G E F B C D A

36. Still problems Random Surfer gets trapped in buckets
Reachable strongly connected component without outlinks
Solution: teleportation matrix E
Matrix of u
.09 I H L M G E F B C D A

37. … finally Google matrix G = aS + (1-a) E Interpretation of G
Where a is the damping factor
Interpretation of G
With probability a, Random Surfer follows a hyperlink from a page (selected at random)
With probability 1-a, Random Surfer jumps to any page (e.g., by entering a new URL in the browser)
PageRank scores are the solution of self-consistent equation
p =pG
=apS + (1-a)u

38. PageRank scores I 1.6 H 1.6 F 3.9 B 38.4 C 34.3 E 8.1 L 1.6 M 1.6 D
3.3 G
1.6

39. Summary
Recursive (or self-consistent) nature of PageRank has roots in social network analysis metrics
PageRank is fundamentally related to random walks on graphs
Lots of research to compute it efficiently
Huge economic and social impact!