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Leting WuXiaowei Ying , Xintao WuAidong Lu and Zhi-Hua ZhouPAKDD 2011Spectral Analysis of k-balanced Signed Graphs1OutlineIntroductionSigned GraphPrevious Study Revisit Spectral Analysis of k-balanced GraphUnbalanced Signed GraphEvaluation2Signed GraphThe original need in anthropology and sociologyLiking(Friend, Trust and etc.) Disliking(Foe, Distrust and etc.) Indifference(Neutrality or No relation)Current needs to analyze real large network data with negative edgesCorrelates of War, Slashdot,Epinion, Wiki Adminship Election and etc.

33using gama as an example to show?? search for previous reportsk-balanced GraphA 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 negativek-balanced graph has no cycle with only one negative edge

4Matrix of Network DataAdjacency 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

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5Revisit: Spectral Coordinate[X.Ying, X.Wu, SDM09]

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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

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OutlineIntroductionSpectral Analysis of k-Balanced GraphBasic ModelModerate Negative Inter-Community EdgesIncrease the Magnitude of Inter-Community Edges Unbalanced Signed GraphEvaluation8Basic ModelLet 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 communities9

Graph with k Disconnected CommunitiesAdjacency Matrix:

First k eigenvectors:

where is the first eigenvector of Spectral Coordinate for node u 10

Moderate Negative Inter-Community Edges11

AB = A + EThe example graph contains two communities following power law random graph model with the size of 600 and 400 nodesModerate Negative Inter-Community EdgesApproximate 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 originNodes with connection to other communities deviate from the k half-lines12

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 Graph13

r1

r2

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 sinceExample: 2-balanced Graph14

r2r1

uExample: 2-balanced GraphCompare 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.

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Add Negative EdgesAdd Positive EdgesIncrease the Magnitude of Inter-Community EdgeIncrease positive inter-community edges:Nodes are pulled closer to each other by the positive edges and finally mixed together.

16p = 1p = 0.3p = 0.1

p: the ratio of inter-community edges to the inner-community edgesIncrease negative inter-community edges:Nodes are pushed further away to the other communities by negative edges;Communities are separable.

Increase the Magnitude of Inter-Community Edges17p = 0.3p = 1p = 0.1

Increase the Magnitude of Inter-Community EdgeTheoretical 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 of18

OutlineIntroductionSpectral Analysis of k-Balanced Graph Unbalanced Signed GraphEvaluation19Unbalanced Signed Graphs20Signed graphs are general unbalanced and can be considered as the result of perturbations on balanced graphs:

A: k disconnected communitiesEin: negative inner community edges -- conflict relation within communitiesEout: positive and negative inter community edges

No Inter-Community Edges:21

Small number of negative edges are added within the communitiesB is still a block diagonal matrixNegative edges would push the nodes towards the negative direction.

With Inter Community Edges22Communities are still separableWhen Ein is moderate and Eout has small number of positive edges, k communities 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

Example: Unbalanced Synthetic Graph23

p=0.1, q=0.1p=0.1, q=0.2With more negative inner-community edges, more nodes are pushed to the negative part of the linesPositive 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 graphOutlineIntroductionSpectral Analysis of K-Balanced GraphUnbalanced Signed GraphEvaluationSynthetic Balanced/Unbalanced GraphsLaplacian Spectral Space

24Synthetic Balanced Graphs253-balanced GraphEven with dense negative inter-community edges, 3 communities are still separablep = 0.1p = 13 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 edgesWith larger q, more nodes are mixed near the origin

Unbalanced Synthetic Graph26p=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 graphUnbalanced Synthetic GraphWith 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=0.2Laplacian Spectral Space28

p=0.1, q=0.2p=1, q=0.2

p=1, q=0The eigenvectors corresponding to the k smallest eigenvalues reflect the community structure, but they are less stable to noise

Compare Adjacency and Laplacian Spectral Spaces29

p=0.1, q=0.2p=1, q=0.2

AdjacencySpectral SpaceLaplacianSpectral SpaceConclusionWe report findings on showing separability of communities in the spectral space of signed adjacency matrixWe 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

30Future WorksEvaluate our findings using various real signed social networks

Develop community partition algorithms and compare with other clustering methods for signed networks

31This work was supported in part by: U.S. National Science Foundation (CCF-1047621, CNS-0831204) for L.Wu, X.Wu, and A.LuJiangsu Science Foundation (BK2008018) and the National Science Foundation of China(61073097) for Z.-H. Zhou

Thank you! Questions?32