1. C OMMUNITIES AND B ALANCE IN S IGNED N ETWORKS : S PECTRAL A PPROACH -Pranay Anchuri*, Malik Magdon Ismail Rensselaer Polytechnic Institute, NY.

2. O UTLINE Introduction Structural Balance Heuristic Spectral Methods Results Conclusion Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

3. S IGNED S OCIAL N ETWORKS Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

5. S TRUCTURAL B ALANCE Stable Unstable Network is strongly balanced if all triads are stable. Notation : Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Positive Edge Negative Edge

6. W EAK S TRUCTURAL B ALANCE Stable Unstable Network is weakly balanced if all triads are stable. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

7. C OMMUNITIES IN B ALANCED N ETWORK Balanced network can be divided so that positive edges lie within communities negative edges between communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

8. Real world networks are rarely structurally balanced. Frustration : Number of edges that disturb the balance. Positive edges between communities + Negative edges within communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

9. Real world networks are rarely structurally balanced. Frustration : Number of edges that disturb the balance. Positive edges between communities + Negative edges within communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Frustration = 1

10. Real world networks are rarely structurally balanced. Frustration : Number of edges that disturb the balance. Positive edges between communities + Negative edges within communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Frustration = 1

11. Real world networks are rarely structurally balanced. Frustration : Number of edges that disturb the balance. Positive edges between communities + Negative edges within communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

12. Community Detection

13. H EURISTIC Ignore the negative edges and cluster the remaining nodes. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

14. H EURISTIC Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

15. H EURISTIC Isolated nodes are added in such a way that minimizes the frustration. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

16. H EURISTIC Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

17. Spectral Methods

18. M INIMIZING F RUSTRATION Community C divided into C1,C2 Positive edges between C1 and C2 increase frustration. Negative edges between C1 and C2 decrease frustration. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

19. M INIMIZING F RUSTRATION Community C divided into C1,C2 Positive edges between C1 and C2 increase frustration. Negative edges between C1 and C2 decrease frustration. C1 C2 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Frustration = 2

20. M INIMIZING F RUSTRATION Community C divided into C1,C2 Positive edges between C1 and C2 increase frustration. Negative edges between C1 and C2 decrease frustration. C1 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute C2 Frustration = 1 Frustration = 2

21. M INIMIZING F RUSTRATION Community C divided into C1,C2 Positive edges between C1 and C2 increase frustration. Negative edges between C1 and C2 decrease frustration. C1 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute C2

22. M ODULARITY Unsigned Modularity : Number of edges within communities – expected number if edges were randomly permuted. Measure of the “surprise” factor. Higher modularity is better. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

23. S IGNED M ODULARITY Signed Modularity Surprise factor due to positive edges within communities and negative edges between communities. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

24. Minimizing Frustration Maximizing Modularity Both objectives reduce to maximizing S T M S Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

25. C OMPUTING THE M AXIMUM Maximizing f (M,S) = S T M S Optimum S : Eigen vector corresponding to maximum Eigen value of M. Eigen vector can be computed by Power Iteration. Requires sparse matrix vector multiplication which is efficient. S ε R n but we need S ε {-1,+1} n !! Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

26. B OOLEAN S OLUTION Rounding : Based on sign of s i, s i >= 0  1 and -1 o/w. Rounding w/ Improvement : Start with an initial Boolean solution and move the nodes one at a time. If there is a sequence of flips such that solution is closer optimum then retain the changes. Complexity : O(N^2). Rounding w/ Partial Improvement: Consider nodes whose magnitude is close to zero. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

27. NodeVal in Eigen Vector 00.55 10.27 20.25 3-0.44 4-0.45 5-0.33 60.10 70.13 8-0.09 3 4 6 7 8 2 0 1 5

28. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute NodeVal in Eigen Vector 00.55 10.27 20.25 3-0.44 4-0.45 5-0.33 60.10 70.13 8-0.09 3 4 6 7 8 2 0 1 5

29. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute NodeVal in Eigen Vector 00.55 10.27 20.25 3-0.44 4-0.45 5-0.33 60.10 70.13 8-0.09 3 4 6 7 8 2 0 1 5

30. M ULTIPLE C OMMUNITIES Communities can be further divided Until frustration cannot be reduced. Modularity cannot be increased. Change in the objective can be reduced to S T M S Also requires sparse matrix vector multiplication. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

31. Results

32. M ODULARITY M AXIMIZATION Algorithm# CommunitiesLargestFrustration (% of –ve edges) Epinions.com Clustering ( 15 means)1569802175.07 Clustering (40 means)4059022195.06 Modularity1556754211.62 Modularity w/ partial improvement 4117072100 Slashdot.com Clustering ( 15 means)1524460259.93 Clustering (40 means)4021666288.69 Modularity1440378176.85 Modularity w/ partial improvement 360031141.72 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Datasets obtained from http://snap.stanford.edu/http://snap.stanford.edu/

33. F RUSTRATION M INIMIZATION Algorithm# CommunitiesLargestFrustration ( % of – ve edges) Epinions.com Two Division27410058.97 Two Division w/ Partial Improvement 27386154.99 Multiple Division206900447.65 Multiple Division w/ Partial Improvement 236899043.92 Slashdot.com Two Division25782466.34 Two Division w/ Partial Improvement 25785364.71 Multiple Division85547962.52 Multiple Division w/ Partial Improvement 105785360.67 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

34. S TRONG VS W EAK B ALANCE Minimum Frustration: = 1 when max # communities =2 = 0 when # communities = 3 ( each node in its own community) Minimum frustration with multiple communities implies weak balance. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

35. N EGATIVE I NCIDENT R ATIO NIR = 3/2 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

36. C ONCLUSION Spectral algorithm to detect communities in signed communities. Objective Functions : Minimizing frustration, Maximizing frustration. Careful assignment of nodes leads to better communities. Structural balance (strong and weak) affects the communities detected. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

37. Thank You Questions ? Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute