Leting Wu Xiaowei Ying, Xintao Wu Aidong Lu and Zhi-Hua Zhou
PAKDD 2011 Spectral Analysis of k-balanced Signed Graphs 1 Slide 2
Outline Introduction Signed Graph Previous Study Revisit Spectral
Analysis of k-balanced Graph Unbalanced Signed Graph Evaluation 2
Slide 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. 3 Slide 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 4 Slide 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 5 Slide 6 Revisit: Spectral Coordinate [X.Ying, X.Wu,
SDM09] 6 Network of US political books sold on Amazon (polbooks,105
nodes, 441 edges) Slide 7 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 Slide 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 8 Slide 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 9
Slide 10 Graph with k Disconnected Communities Adjacency Matrix:
First k eigenvectors: where is the first eigenvector of Spectral
Coordinate for node u 10 Slide 11 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 Slide 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:
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
Slide 13 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 Slide 14 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 Slide 15 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 Slide 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. 16 p = 1 p = 0.3 p = 0.1
p: the ratio of inter-community edges to the inner-community edges
Slide 17 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 Slide 18 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
Slide 19 Outline Introduction Spectral Analysis of k-Balanced Graph
Unbalanced Signed Graph Evaluation 19 Slide 20 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 Slide 21 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. Slide 22 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 Slide 23 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 Slide 24 Outline Introduction
Spectral Analysis of K-Balanced Graph Unbalanced Signed Graph
Evaluation Synthetic Balanced/Unbalanced Graphs Laplacian Spectral
Space 24 Slide 25 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 Slide 26 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 Slide 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) 27 p=1,
q=0.2 Slide 28 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 Slide 29 Compare Adjacency and Laplacian Spectral
Spaces 29 p=0.1, q=0.2p=1, q=0.2 Adjacency Spectral Space Laplacian
Spectral Space Slide 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 30 Slide 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 31 Slide 32 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 Thank you! Questions? 32
