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

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

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. 3

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 4

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 5

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

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

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 8

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 9

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

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

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: 1. Nodes without connection to other communities lie on k quasi-orthogonal half-lines starting from the origin 2. Nodes with connection to other communities deviate from the k half-lines 12

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. Example: 2-balanced Graph 13 r1r1 r2r2

The spectral coordinate of node 1. When u does not connect to, u lies on line since 2. When u connects to, spectral coordinate of node u and has an obtuse angle since Example: 2-balanced Graph 14 r2r2 r1r1 u

Example: 2-balanced Graph Compare with adding positive edges when is small: 1. Two half-lines exhibit a clockwise rotation from axes. 2. Spectral coordinates of node u and has an acute angle. 15 Add Negative EdgesAdd Positive Edges

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. 16 p = 1 p = 0.3 p = 0.1 p: the ratio of inter-community edges to the inner-community edges

Increase negative inter-community edges: 1. Nodes are “pushed” further away to the other communities by negative edges; 2. Communities are separable. Increase the Magnitude of Inter-Community Edges 17 p = 0.3p = 1p = 0.1

Increase the Magnitude of Inter-Community Edge Theoretical explanation 1. R is an approximately orthogonal transformation. 2. The direction of deviation is specified by E: When inter-communities edges are all negative, the deviation of is towards the negative direction of 18

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

Unbalanced Signed Graphs 20 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

No Inter-Community Edges: 21 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.

With Inter Community Edges 22 Communities are still separable 1. When E in is moderate and E out has small number of positive edges, k communities still form k quasi- orthogonal lines; nodes with inter community edges deviate from the k lines. 2. Some nodes lie on the negative part of the k lines. 3. Rotation effect and node deviation are comprehensive results affected by the sign of edges and the signs of the end points

Example: Unbalanced Synthetic Graph 23 p=0.1, q=0.1p=0.1, q=0.2 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: the ratio of inter-community edges to the inner-community edges q: the ratio of flipped edges from a balanced graph

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

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

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 Unbalanced Synthetic Graph 26 p=0.1, q=0.1p=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

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) 27 p=1, q=0.2

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

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

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 30

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

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