1. Presentation: Random Walk Betweenness, J. Govorčin Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011 Random Walk Betweenness Presented by Jelena Govorčin

2. Presentation: Random Walk Betweenness, J. Govorčin Introduction to Centrality of a graph node Random walk betweenness – Basic matrix notation – Calculation of random walks betweenness Examples and applications – Simple graph example – Correlation with other measures – Example applications Conclusions Overview Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

3. Presentation: Random Walk Betweenness, J. Govorčin Importance of vertices in a graph – Roles played by people on social network – Communication stability of a network – Epidemiological application, etc. Degree – number of edges incudent on a vertex in a network Closeness – mean geodesic distance between a vertex and all other vertices reachable from it Betweenness (measure of the extent to which a vertex lies on the paths between others) – Shortest-path betweenness – Flow betweenness – Random walk betweenness Centrality measures Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

4. Presentation: Random Walk Betweenness, J. Govorčin Degree The first, and simplest centrality measure. INTERPRETATION: Immediate risk of node for catching whatever is flowing through the network (such as a virus, or some information). For a graph G=(V,E) with n vertices, the degree centrality C D (v) for vertex v is: C D (v) = deg(v) The number of ties an actor has in social network. A measure, in some sense, of the popularity of an actor. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

5. Mean geodesic distance between a vertex and all other vertices reachable from it. INTERPRETATION: How long it will take information to spread from a given vertex to others in the network? For a graph G=(V,E) with n vertices, the closeness centrality C C (v) for vertex v is: Presentation: Random Walk Betweenness, J. Govorčin Closeness Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

6. Presentation: Random Walk Betweenness, J. Govorčin Betweenness (Freeman,1977) Measure of the extent to which a vertex lies on the paths between others. Shortest-path betweenness of a vertex v is defined to be the fraction of shortest paths between pairs of vertices in a network that pass through v. - the number of shortest paths from s to t that pass through a vertex v - the number of shortest paths from s to t Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

7. Presentation: Random Walk Betweenness, J. Govorčin Example of Shortest-path Betweenness Vertices A and B have high (shortest-path) betweenness in this configuration, while vertex C does not. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

8. Does information flow only along geodesic paths? News, rumor, fad, message – does it know the ideal route? To get from one place to another more likely a message wanders around more randomly, encountering who it will. A realistic betweenness measure should include non-geodesic paths in addition to geodesic ones! Presentation: Random Walk Betweenness, J. Govorčin Drawbacks of Shortest-path Betweenness Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

9. Presentation: Random Walk Betweenness, J. Govorčin Flow Betweenness (Freeman, 1991) Includes contributions for some non-geodesic paths. Based on the idea of maximum flow : What is the maximum possible flow between a given source s and target t? DEFINITION: The amount of flow through vertex v when the maximum flow is transmitted from s to t, averaged over all s and t. INTERPRETATION: Measure of betweenness of vertices in a network in which a maximal amount of information is continuously pumped between all sources and targets. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

10. Presentation: Random Walk Betweenness, J. Govorčin Example of Flow Betweenness In calculations of flow betweenness, vertices A and B in this configuration get high scores while vertex C not. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

11. Presentation: Random Walk Betweenness, J. Govorčin Drawbacks of Flow Betweenness Does information “know” the ideal route from each source to each target, in order to realize the maximum flow? Unrealistic for many practical situations. It is often the case that flow does not take any sort of ideal path from source to target (the shortest path, the maximum flow path, or another kind of ideal path). Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

12. Presentation: Random Walk Betweenness, J. Govorčin Basic Matrix Notations Adjacency Matrix A Degree Matrix D Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

13. Presentation: Random Walk Betweenness, J. Govorčin Random Walk Betweenness Random-walk betweenness of a vertex i is equal to the (“net”) number of times that a random walk starting at s and ending at t passes through i along the way, averaged over all s and t. This measure is appropriate to a network in which information wanders about essentially at random until it finds its target. Includes contributions from many paths that are not optimal in any sense. Random-walk betweenness and the shortest-path betweenness are at opposite ends of a spectrum of possibilities. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

14. We are considering absorbing random walk Suppose that the current state of this walk is vertex j, then is the probability the next state is vertex i. Matrix notation: Exception: - we can remove row and column t without affecting transitions between any other vertices. Presentation: Random Walk Betweenness, J. Govorčin Calculating Random Walk Betweenness 1 Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

15. New notation (after removing row and column t) - for a walk starting in s, the probability that after r steps the current state will be j. - the probability that we then take a step to an adjacent vertex i. Summing over all values of r: Presentation: Random Walk Betweenness, J. Govorčin Calculating Random Walk Betweenness 2 Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

16. Matrix notation: where s is a source vector with elements - the net flow of the random walk along the edge from j to i Presentation: Random Walk Betweenness, J. Govorčin Calculating Random Walk Betweenness 3 Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

17. The net flow (of random walk starting at s and ending at t) through vertex i (a half the sum of the flows on the incident edges) The final net flow of random walks through vertex i Presentation: Random Walk Betweenness, J. Govorčin Calculating Random Walk Betweenness 4 Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

18. Presentation: Random Walk Betweenness, J. Govorčin Simple Graph Examples Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

19. Presentation: Random Walk Betweenness, J. Govorčin Correlation With Other Measures 1 Moderately highly correlated with degree Very highly correlated with shortest-path b. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

20. Presentation: Random Walk Betweenness, J. Govorčin Correlation With Other Measures 2 The size of the vertices indicates their random walk betweenness. The pointed shaded vertices are those for which the random-walk betweenness is much greater than shortest-path betweenness (twice or more). The largest network component of a sexual contacts of high-risk actors in Colorado Springs, CO (Potterat et al. 2002). Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

21. Presentation: Random Walk Betweenness, J. Govorčin Network Example 1 The network of intermarriage relations between the 15th century Florentine families (Padgett and Ansell, 1993). Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

22. Presentation: Random Walk Betweenness, J. Govorčin Network Example 2 The largest component of the co-authorship network of scientists working on networks (Newman and Park, 2003). Vertices on a single path from one part of the network to another (labeled “A”) get a high score. Vertices labeled “B” get a high score to, even though they lie on one of several paths between different parts of the network. Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

23. Presentation: Random Walk Betweenness, J. Govorčin The measure of betweenness based on random walks counts all paths between vertices and makes no assumptions of optimality. Useful for finding vertices of high centrality that do not happen to lie on geodesic pathr or on the paths formed by max-flow cut-sets. It correlates to other measures of centrality and gives more realistic scores to vertices in mentioned applications. It can be calculated using matrix inversion in O(n 3 ) (computationally tractable). Conclusions Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

24. Presentation: Random Walk Betweenness, J. Govorčin 1.Freeman, L. C., Centrality in social networks: Conceptual clarification, Social Networks 1, 215-239, 1979. 2.Newman, M. E. J., A measure of betweenness centrality based on random walks, Social Networks 27, 39-54, 2005. 3.Noh, J. D. and Rieger, H., Random walk on complex networks, Phys. Rev. Lett., 92 (11), 2004. References Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011

25. Presentation: Random Walk Betweenness, J. Govorčin Laboratory for Data Technologies, Faculty of Information Studies, Novo mesto – September 22, 2011 Thank you for your attention!