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Signed Network Communities: A Spectral Perspective

Explore the structural balance and heuristic approach in signed networks using spectral methods, with a focus on frustration and community detection. Real-world network examples are analyzed, emphasizing the challenges and strategies for minimizing frustration while maximizing modularity. Learn about computing the maximum eigenvalue and rounding methods for robust solutions.

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Signed Network Communities: A Spectral Perspective

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  1. Communities and Balance in Signed Networks : Spectral Approach -Pranay Anchuri*, MalikMagdon Ismail Rensselaer Polytechnic Institute, NY.

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

  3. Signed Social Networks Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  4. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  5. Structural Balance • Stable • Unstable • Network is strongly balanced if all triads are stable. • Notation : Positive Edge Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Negative Edge

  6. Weak Structural Balance • Stable • Unstable • Network is weakly balanced if all triads are stable. Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  7. Communities in Balanced Network • 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 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

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

  14. Heuristic Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

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

  16. Heuristic Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  17. Spectral Methods Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  18. Minimizing Frustration • 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. Minimizing Frustration • 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 C2 C1 Frustration = 2 Frustration = 2

  20. Minimizing Frustration • 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 C1 C2 Frustration = 2 Frustration = 1

  21. Minimizing Frustration • 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 C1 C2

  22. Modularity • 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. Signed Modularity • 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 ST M S Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  25. Computing the Maximum • Maximizing f(M,S) = ST 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 Rn but we need S ε {-1,+1}n !! Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  26. Boolean Solution • Rounding : • Based on sign of si, si >= 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. 5 7 6 4 3 8 0 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute 2 1

  28. 5 7 6 4 3 8 0 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute 2 1

  29. 5 7 6 4 3 8 0 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute 2 1

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

  31. Results Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  32. Modularity Maximization Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute Datasets obtained from http://snap.stanford.edu/

  33. Frustration Minimization Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  34. Strong vs Weak Balance • 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. Negative Incident Ratio NIR = 3/2 Pranay Anchuri, Malik Magdon Ismail, Rensselaer Polytechnic Institute

  36. Conclusion • 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

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