Spectral Analysis of k -balanced Signed Graphs

1. Spectral Analysis of k-balanced Signed Graphs Leting Wu Xiaowei Ying , Xintao Wu Aidong Lu and Zhi-Hua Zhou PAKDD 2011

2. Outline • Introduction • Signed Graph • Previous Study Revisit • Spectral Analysis of k-balanced Graph • Unbalanced Signed Graph • Evaluation

3. Signed Graph • The original need in anthropology and sociology • Liking(Friend, Trust and etc.) • Disliking(Foe, Distrust and etc.) • Indifference(Neutrality or No relation) • Current needs to analyze real large network data with negative edges • Correlates of War, Slashdot,Epinion, Wiki Adminship Election and etc.

4. k-balanced Graph • A graph is a k-balanced graph if the node set can be divided into k disjoint subsets, edges connecting any two nodes from the same subset are all positive and edges connecting any two nodes from the different subsets are all negative • k-balanced graph has no cycle with only one negative edge

5. Matrix of Network Data • Adjacency Matrix A (symmetric) • Adjacency Eigenpairs • The k-dimensional subspace spanning by the first k eigenvectors reflects most topological information of the original graph for certain k

6. Revisit: Spectral Coordinate [X.Ying, X.Wu, SDM09] • Network of US political books • sold on Amazon • (polbooks,105 nodes, 441 edges)

7. Revisit: Line Orthogonality [L.Wuet al., ICJAI11] • With sparse inter-community edges, the spectral coordinate for node u is approximated by:where is the ithrow of

8. Outline • Introduction • Spectral Analysis of k-Balanced Graph • Basic Model • Moderate Negative Inter-Community Edges • Increase the Magnitude of Inter-Community Edges • Unbalanced Signed Graph • Evaluation

9. Basic Model • Let B be the adjacency matrix of a k-balanced graph: A: the adjacency matrix of a graph with k disconnected communities E: the negative edges across communities

10. Graph with k Disconnected Communities • Adjacency Matrix: • First k eigenvectors: where is the first eigenvector of • Spectral Coordinate for node u

11. Moderate Negative Inter-Community Edges A B = A + E The example graph contains two communities following power law random graph model with the size of 600 and 400 nodes

12. Moderate Negative Inter-Community Edges • Approximate spectral coordinates of B by A and E. When E is non-positive and its 2-norm is small: • Properties of the spectral coordinates for B: • Nodes without connection to other communities lie on k quasi-orthogonal half-lines starting from the origin • Nodes with connection to other communities deviate from the k half-lines

13. Example: 2-balanced Graph • Two quasi-orthogonal lines: • The direction of the rotation when E is non-positive: So two half lines rotate counter-clockwise. We also notice so the two lines are approximately orthogonal. r2 r1

14. Example: 2-balanced Graph • The spectral coordinate of node • When u does not connect to , u lies on line since • When u connects to , spectral coordinate of node u and has an obtuse angle since r2 r1 u

15. Example: 2-balanced Graph • Compare with adding positive edges when is small: • Two half-lines exhibit a clockwise rotation from axes. • Spectral coordinates of node u and has an acute angle. Add Negative Edges Add Positive Edges

16. Increase the Magnitude of Inter-Community Edge • Increase positive inter-community edges: Nodes are “pulled” closer to each other by the positive edges and finally mixed together. p = 0.3 p = 0.1 p = 1 • p: the ratio of inter-community edges to the inner-community edges

17. Increase the Magnitude of Inter-Community Edges • Increase negative inter-community edges: • Nodes are “pushed” further away to the other communities by negative edges; • Communities are separable. p = 0.1 p = 0.3 p = 1

18. Increase the Magnitude of Inter-Community Edge • Theoretical explanation • R is an approximately orthogonal transformation. • The direction of deviation is specified by E: When inter-communities edges are all negative, the deviation of is towards the negative direction of

19. Outline • Introduction • Spectral Analysis of k-Balanced Graph • Unbalanced Signed Graph • Evaluation

20. Unbalanced Signed Graphs • Signed graphs are general unbalanced and can be considered as the result of perturbations on balanced graphs: A: k disconnected communities Ein: negative inner community edges -- conflict relation within communities Eout: positive and negative inter community edges

21. No Inter-Community Edges: • Small number of negative edges are added within the communities • B is still a block diagonal matrix • Negative edges would “push” the nodes towards the negative direction.

22. With Inter Community Edges • Communities are still separable • When Ein is moderate and Eouthas small number of positive edges, kcommunities still form k quasi-orthogonal lines; nodes with inter community edges deviate from the k lines. • Some nodes lie on the negative part of the k lines. • Rotation effect and node deviation are comprehensive results affected by the sign of edges and the signs of the end points

23. Example: Unbalanced Synthetic Graph • With more negative inner-community edges, more nodes are “pushed” to the negative part of the lines • Positive inter-community edges eliminate some rotation effect of the lines caused by negative inter-community edges p=0.1, q=0.1 p=0.1, q=0.2 p: the ratio of inter-community edges to the inner-community edges q: the ratio of flipped edges from a balanced graph

24. Outline • Introduction • Spectral Analysis of K-Balanced Graph • Unbalanced Signed Graph • Evaluation • Synthetic Balanced/Unbalanced Graphs • Laplacian Spectral Space

25. Synthetic Balanced Graphs • 3-balanced Graph • Even with dense negative inter-community edges, 3 communities are still separable 3 disconnected communities of power law degree distribution with 600/500/400 nodes p = 0.1 p = 1 • p: the ratio of inter-community edges to the inner-community edges

26. Unbalanced Synthetic Graph • Even the graph is unbalanced, nodes from the three communities exhibit three lines starting from the origin and some nodes deviate from the lines due to inter-community edges • With larger q, more nodes are mixed near the origin p=0.1, q=0.1 p=0.1, q=0.2 p: the ratio of inter-community edges to the inner-community edges q: the ratio of flipped edges from a balanced graph

27. Unbalanced Synthetic Graph • With dense inter-community edges, we can not observe the line pattern diminishes but nodes from 3 communities are separable (p=1) p=1, q=0.2

28. Laplacian Spectral Space The eigenvectors corresponding to the k smallest eigenvalues reflect the community structure, but they are less stable to noise p=1, q=0 p=0.1, q=0.2 p=1, q=0.2

29. Compare Adjacency and Laplacian Spectral Spaces Adjacency Spectral Space Laplacian Spectral Space p=0.1, q=0.2 p=1, q=0.2

30. Conclusion • We report findings on showing separability of communities in the spectral space of signed adjacency matrix • We find communities in a k-balanced graph are distinguishable in the spectral space of signed adjacency matrix even if connections between communities are dense • We conduct the similar analysis over Laplacian Spectral Space and find it is not suitable to analyze unbalanced graphs

31. Future Works • Evaluate our findings using various real signed social networks • Develop community partition algorithms and compare with other clustering methods for signed networks

32. Thank you! Questions? This work was supported in part by: U.S. National Science Foundation (CCF-1047621, CNS-0831204) for L.Wu, X.Wu, and A.Lu Jiangsu Science Foundation (BK2008018) and the National Science Foundation of China(61073097) for Z.-H. Zhou