1. A framework For Community Identification in Dynamic Social Networks Chayant, Tanya Berger-Wolf, David Kempe [KDD’07] Advisor ： Dr. Koh Jia-Ling Advisor ： Dr. Koh Jia-Ling Speaker ： Che-Wei Liang Date ： 2008.1.8

2. Outline Introduction Problem formulation Finding optimal colorings Group Coloring Heuristics Experiment Conclusion

3. Introduction Social networks – Graphs of interactions between individuals.

4. Introduction What is Community? – collections of individuals who interact unusually frequently. – reveal interesting properties shared by member, such as common hobbies, occupations. Why dynamic community? – may have more interesting properties.

5. History of Interactions t=1 1 2 3 4 5 Assume discrete time and interactions in form of complete subgraphs. Aggregated network 5 4 2 3 1 2 32 1 1 1

6. Approach: Graph Model 5 5 5 5 5 1234 1234 1234 1234 1234 t=1 t=2 t=3 t=4 t=5 12345 54 1 23 52341 5234 5241

7. Preliminaries

8. Problem Formulation Behavior of individuals assumption: – Individuals and groups represent exactly one community at a time. – Concurrent groups represent distinct communities.

9. Problem Formulation Behavior of individuals assumption (cont.): – Conservatism: community affiliation changes are rare. – Group Loyalty: individuals observed in a group belong to the same community. – Parsimony: few affiliations overall for each individual.

10. Approach: Color = Community Valid coloring: distinct color of groups in each time step

11. i-cost Conservatism: switching cost ( α ) Group loyalty: Being absent ( β 1) Being different ( β 2) Parsimony: number of colors ( γ )

12. g-cost Conservatism: switching cost ( α ) Group loyalty:  Being absent ( β 1)  Being different ( β 2) Parsimony: number of colors ( γ )

13. c-cost Conservatism: switching cost ( α ) Group loyalty:  Being absent ( β 1)  Being different ( β 2) Parsimony: number of colors ( γ )

14. Minimum Community Interpretation For a given cost setting, (α,β1,β2,γ), find vertex coloring that minimizes total cost. – Color of group vertices = Community structure – Color of individual vertices = Affiliation sequences Problem is NP-Complete and APX-Hard

15. Finding Optimal Colorings Individual Coloring – G(t, x): g-cost of coloring i at time step t with color x – I(t, x, y): i-cost of coloring I at time steps t and i-1 with colors x and y. – C(x, R): c-cost of using color x when R is the set of colors used in prior steps.

16. Finding Optimal Colorings Group Coloring – Using exhaustive search over all group colorings. – Speed up by Branch-and-Bound techniques.

17. Group Coloring Heuristics Bipartite Matching Heuristic – Using standard flow techniques. Greedy Heuristics – Maximize “similarity” – Jaccard’s index: Jac(g, g’) = – Repeatedly select the pair(g, g’) with highest similarity, decide g, g’ should have same color.

18. Experiment Synthetic Data sets

19. Southern Women Data Set by Davis, Gardner, and Gardner, 1941 Photograph by Ben Shaln, Natchez, MS, October; 1935 Aggregated network Event participation

20. An Optimal Coloring: ( α, β 1, β 2, γ )=(1,1,3,1) Core Periphery Core

21. An Optimal Coloring: ( α, β 1, β 2, γ )=(1,1,1,1) Core Periphery Core

22. Conclusion An optimization-based framework for finding communities in dynamic social networks. Finding an optimal solution is NP-Complete and APX-Hard. Model evaluation by exhaustive search. Heuristic algorithms for larger data sets. Heuristic results comparable to optimal.