Clustering Social Networks

1. Clustering Social Networks Isabelle Stanton, University of Virginia Master of Science Thesis Defense

2. Outline • Motivation • Previous Work • Finding Tightly Knit Clusters • Finding Loosely Knit Clusters • Group Recommendations • Future Work

3. Motivation • Many large social networks: • A fundamental problem is finding communities automatically • Viral and Targeted Marketing • Recommendation Engines

4. Previous Work – Spectral Methods • Cuts the graph based on an eigenvector • Spectral Methods: • Kannan, Vempala, Vetta 2000, Spielman and Teng 1996, Shi and Malik 2000, Kempe and McSherry 2004, Karypis and Kumar 1998 and many others • cut = partitioning of all elements

5. Communities in Social Networks • Disjoint partitionings are not good for social networks

6. Objective: Internal Density,  Each vertex in C is adjacent to at least  fraction of (the rest of) C Examples: =1/2 =3/4 =1

7. Objective: External Sparsity,  Each vertex outside of C is adjacent to at most  of C  <  =1/5, =1 =1

8. (α, β)-Clusters • C is an (α, β)- cluster if: • Internally Dense: Every vertex in the cluster neighbors at least a β fraction of the cluster • Externally Sparse: Every vertex outside the cluster neighbors at most an α fraction of the cluster (1/4, 2/3) (1/4, 1)

9. Contributions of this work • Definition of criterion • Combinatorial results • 3 overlap results • Bound on number of (α,1) clusters • Three algorithms for varying cases • Experiments validating assumptions on real social networks • Novel formulation of group recommendation problem with experiments

10. Previous Work – (α, β)-clusters • Solved Areas: Our Contributions: 1 (1- ε,1) – Tsukiyama et al, Johnson et al. α = 0 – connected components α β > ½ + α/2 – Algorithm 1 and 2 0 α < β3 – Algorithm 3 1 0 β

11. Outline • Motivation • Previous Work • Finding Tightly Knit Clusters • Finding Loosely Knit Clusters • Group Recommendations • Future Work

12. Too Many Clusters.. n vertices MISSING edges drawn x1 y1 x2 y2 ... xn/2 yn/2 Problem:Every vertex in every cluster has as many neighbors outside the cluster as in it

13. ρ-Champions Ben Stiller Gwenyth Paltrow Will Ferrell Vince Vaughn Wes Anderson Owen Wilson ρ-champion Steve Martin Bill Murray Anjelica Houston

14. ρ-Champions • Def: A vertex is a ρ-champion of C if it has at most ρ|C| neighbors outside C • Claim: If ρ < 2β – 1 – α, every vertex can ρ-champion at most one cluster

15. Outline • Motivation • Previous Work • Combinatorial properties • Finding Tightly Knit Clusters • Deterministic Algorithm • Finding Loosely Knit Clusters • Group Recommendations • Future Work

16. Let c be a ρ-champion If v in C, then v and c share at least (2β -1)|C| neighbors If v is outside C then v and c share at most (ρ + α)|C| neighbors Intuition behind the Algorithm v α|C| β|C| v c ρ|C| β|C| (2β-1)|C| c

17. Deterministic Algorithm • To find all clusters of size s: • for each c in V do • C←  • For each v within two steps of c do • If v and c share (2β – 1)s neighbors then add v to C • If C is an (α, β)-cluster then output C

18. Algorithmic Guarantees • Claim: Our algorithm will find all clusters of size s where β > ½ + (ρ + α)/2 • Runs in O(d0.7n1.9+n2+o(1)) time where d is the average degree • d is a small constant for social networks so O(n2)

19. Evaluation • Do ρ-champions exist in real graphs? • Tsukiyama’s algorithm finds all maximal cliques ((1-ε, 1)-clusters) in a graph • We compare our algorithm’s output with Tsukiyama’s ground truth

20. Theory Co-Author Dataset Results • Found 797 of 854 clusters ~ 93%

21. Outline • Motivation • Previous Work • Combinatorial properties • Finding Tightly Knit Clusters • Finding Loosely Knit Clusters • Technical Challenges • Randomized Algorithm • Group Recommendations • Future Work

22. Loosely Knit Clusters • β≤ ½ • Technical Problem: (0, 1/2)

23. Connectivity Assumption • Every subset of a cluster has an outside vertex in the cluster that neighbors more than a β-fraction Does satisfy assumption! (β = 2/7) Doesn’t satisfy assumption

24. Loosely Knit Randomized Algorithm • α < β3 • Two phases • Phase 1: • Draw a sample of the ρ-champion’s neighbors • Sample neighbors to add to the seed • Stop when the seed is “big enough” • Phase 2: • Exploit connectivity assumption to deterministically grow the seed into the cluster

25. Example • Phase 1: • Sample of the ρ-champion’s neighbors • Sample neighbors to add to the sample • Stop when the sample is “big enough” • Phase 2 • Deterministically grow cluster

26. Example • Phase 1: • Sample of the ρ-champion’s neighbors • Sample neighbors to add to the sample • Stop when the sample is “big enough” • Phase 2 • Deterministically grow cluster

27. Example • Phase 1: • Sample of the ρ-champion’s neighbors • Sample neighbors to add to the sample • Stop when the sample is “big enough” • Phase 2 • Deterministically grow cluster

28. Example • Phase 1: • Sample of the ρ-champion’s neighbors • Sample neighbors to add to the sample • Stop when the sample is “big enough” • Phase 2 • Deterministically grow cluster

29. Why does this work? • Random sampling guarantees the expected number of neighbors an outside vertex has with the seed is small • The connectivity assumption guarantees we’ll always make progress • Guarantees: Finds all clusters where α < β3 with probability 1 – δ • Runs in time O(n3/δ log(n/δ) |C|2)

30. Outline • Motivation • Previous Work • Combinatorial properties • Finding Tightly Knit Clusters • Finding Loosely Knit Clusters • Group Recommendations • Future Work

31. Group Recommendations • Clustering isn’t the end goal • What can we do with (α,β)-clusters? • We built a group recommendation engine powered by our clusters • Recommended groups to users of Orkut and LiveJournal

32. Recommendation Model • Hofmann and Puzicha ‘99 5 5/60 25/60 20 2/3 25 60 people 3/4 1/3 15 10 1/2 10 20 people

33. Previous Work • Kleinberg and Sandler: Given the Groups x User matrix, use matrix decomposition • Their code works ~ 100K variables max • No one uses the friendship graph or clusters!

34. Experimental Setup • Hold out 10% of users with group memberships • Cluster the rest • Create recommendations for held out users based on clusters

35. Results – LiveJournal Dataset Held out: 355,495 users – Succeeded on: 210,455

36. Results – LiveJournal Dataset

37. Conclusions • Defined (α, β)-clusters • Focus: Overlapping clusters • Introduced ρ-champions • Developed algorithms for a subset of the problem • Ran experiments to validate assumptions and show utility of the clusters • Introduced new interpretation of the recommendation model

38. Future Work • Algorithms that reduce the necessary α-β gap • Relaxing ρ-champion restriction • Weighted and directed graphs • Decentralized algorithms • Streaming algorithms • Expanding work on group recommendations

39. Citations • Clustering Social Networks, N. Mishra, R. Schreiber, I. Stanton and R. E. Tarjan, The 5th Workshop on Algorithms and Models for the Web-Graph, WAW2007. LNCS, vol 4863, pp. 56-67. • Clustering Social Networks,N.Mishra, R. Schreiber, I. Stanton and R. E. Tarjan, Journal of Internet Mathematics (under submission)

40. NewKid Algorithm • Input: Graph, Groups, (α,β)-clusters • For each group g and cluster c: • P(g|c) = |members of c in g| / | members in c| • For each new kid, u: • P(c|u) = |friends of u in c| / |friends of u| • Recommend g that maximizes Σc p(g|c)P(c|u)

41. Results – Orkut Dataset

42. Results – Orkut Dataset

43. HEP Co-Author Dataset Results • Found 115 of 126 clusters ~ 90%

44. LiveJournal Dataset Results • Too big to run Tsukiyama. Found 4289 clusters, 876 have large ρ-champions

45. Datasets • High Energy Physics Co-Authorship Graph • Theory Co-authorship graph • A subset of LiveJournal.com τ(v) = the neighbors and neighbors’ neighbors of v

46. Randomized Algorithm • To find all (α, β)-clusters of size s: • for each c in V do: • Repeat k times: • Draw a random sample S of size t from c’s neighbors • C← S U {c} • For each v within two steps of c do • If v has (2β – 1)/ β t neighbors in S then add v to C • If C is an (α, β)-cluster then output C

47. Randomized Algorithm • t = O( log(n / δ) ), k = O( n / δ ) • Guarantees: Finds all clusters where α < 2β – 1 with probability 1 – δ • Runs in time O(n3/δ log(n/δ) (log(n/δ)+|C|)) • Worst case: O(n4/δ log(n/δ)) • Average case: O(n2/δ log2(n/δ) d2)

48. Combinatorial Properties - Overlaps • Let A and B be (α, β)-clusters with |A|=|B| • Theorem: A and B overlap by at most (1-(β-α))|A| vertices 1 0 0 1

49. Combinatorial Properties - |Clusters| • Claim: There are at most (α,1)-clusters of size s in a graph • Bound is tight as α→ 1 and α = 0. Seems loose elsewhere • Proof is from Steiner Systems • 7 points, block size = 3, restriction = 2 • {1,2,4},{2,3,5},{3,4,6},{4,5,7},{1,5,6},{2,6,7},{1,3,7}

50. Outline • Motivation • Previous Work • Combinatorial properties • Finding Tightly Knit Clusters • Finding Loosely Knit Clusters • Experiments and Group Recommendations • Are ρ-champions valid? • What are these clusters good for? • Future Work