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Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization
Jérôme Kunegis¹, Stephan Schmidt¹, Andreas Lommatzsch¹ & Jürgen Lerner²¹DAI Lab, Technische Universität Berlin, ²Universität Konstanz, Germany
10th SIAM International Conference on Data Mining, April 29–May 1, Columbus, Ohio
Kunegis et al. Spectral Analysis of Signed Graphs 2
Introduction: Negative Edges
Some websites allow you to have foes :
Example: Slashdot Zoo (Kunegis 2009)
Kunegis et al. Spectral Analysis of Signed Graphs 3
Introduction: Signed Graphs
• The resulting social network is signed
• Edges are positive or negative
• In this talk: we use the graph Laplacian to study signed graphs
Example: Slashdot Zoo (Kunegis 2009)
me
Friend ofFoe of
Fan ofFreak of
Kunegis et al. Spectral Analysis of Signed Graphs 4
Outline
Introduction: Signed Graphs
1. Negative Edges and the Laplacian2. Balance, Conflict and the Graph Spectrum3. Communities, Cuts and Clustering4. Resistance, Conductivity and Link Prediction
Discussion
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1. Negative Edges and the Laplacian
� Graph drawing: Place each node at the center of its neighbors
v0 = (1/3) (v1 + v2 + v3)
Algebraically: D v = A v
Solution 1: Upper eigenvectors of D−1 A using A = {0, 1}n×n
Solution 2: Lower eigenvectors of D – A and Dii = Σj Aij
We look at solution 2: L = D − A is the Laplacian matrix
v0
v1
v2 v3
Kunegis et al. Spectral Analysis of Signed Graphs 6
Drawing Signed Graphs
• Replace ‘negative’ neighbors by their antipodal points
v0 = (1/3) (−v1 + v2 + v3)
Solution: lower eigenvectors of L = D − A
Using A = {0, −1, +1}n×n
And Dii = Σj | Aij|
v0
v1
v2v3
−v1
Kunegis et al. Spectral Analysis of Signed Graphs 7
Example: Synthetic Graph
Unsigned Graph Drawing → Signed Graph Drawing
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2. Balance, Conflict and the Graph Spectrum
• ‘Balanced’ graphs have a perfect 2-clustering
• Invert all negative edges
• Effect on the Laplacian decomposition: Inversion of all eigenvectors of one cluster
• Therefore: The spectrum of a balanced graph is the same as for the underlying unsigned graph (λ₁ = 0)
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The Laplacian Spectrum of Unbalanced Graphs
• Networks with conflict contain odd cycles
• The Laplacian is always positive semidefinite
xTLx = Σij |Aij|(xi − sgn(Aij) xj)² ≥ 0
• In unbalanced networks: λ₁ > 0
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Algebraic Conflict
• λ₁ denotes conflict
Network λ₁₁₁₁
MovieLens 100k 0.4285
MovieLens 1M 0.3761
Jester 0.06515
MovieLens 10M 0.006183
Slashdot Zoo 0.006183
Epinions 0.004438
Conflict
For effect of size, see Appendix
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3. Communities, Cuts and Clustering
The tribal groups of the Eastern Central Highlands of New Guinea can be friends (‘rova’) or enemies (‘hina’)
Graphic uses two lower eigenvectors of L
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Finding Communities
The Laplacian matrix finds communities :
• Communities are connected by many positive edges
• Community are separated by many negative edges
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Signed Spectral Clustering
• Compute the d lower eigenvectors of L• Use k-means to cluster nodes in this d-dimensional space
• Minimize signed normalized cut between communities X and Y
SNC(X, Y) = (|X|−1 + |Y|−1) · (2 pos(X, Y) + neg(X, X) + neg(Y, Y))
pos/neg: number of positive/negative edges between communities
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Example: Wikipedia Reverts
• Users revert users on controversial Wikipedia article ‘Criticism of Prem Rawat’
• All edges are negative
• Distance to center normalized to unit
• Four clusters are apparent
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4. Resistance, Conductivity and Link Prediction
• Consider a network of electrical resistances:
• Between any two nodes, the network has an effective resistance
• The resistance distance is a squared Euclidean metric
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Link Prediction
• The resistance distance can be used for link prediction :
– Long paths count less– Parallel paths count more
dist(i,j) = (L+)ii + (L+)jj − (L+)ij − (L+)ji
• Problem: How to handle negative edges ?
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Voltage Inversion
• Solution: inverting amplifier
dist(i,j) = (L+)ii + (L+)jj − (L+)ij − (L+)ji
• Using signed Laplacian L• Is squared Euclidean because L is positive semidefinite
−ww −
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• Task: Predict the sign of new links
• Problem: Find a function F(A) = B
Evaluation: Link Sign Prediction
Known positive links (A)
Links to be predicted (B)
Known negative links (A)
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Graph Kernels
Link prediction functions using the Laplacian:
• L+ – Signed Laplacian kernel
• (I + αL)−1 – Signed regularized Laplacian kernel
• exp(−αL) – Signed ‘heat diffusion’
Other link prediction functions:
• (A)k – Rank reduction
• exp(A) – Matrix exponential
• Poly(A) – Path counting
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Evaluation Results
• MovieLens: predict good / bad rating
• Best rating prediction: signed regularized Laplacian graph kernel
Link Prediction RMSE
Rank reduction 0.838
Path counting 0.840
Matrix exponential 0.839
Signed resistance distance 0.812
Signed regularized Laplacian 0.778
Signed heat diffusion 0.789
Kunegis et al. Spectral Analysis of Signed Graphs 21
Summary
� The Laplacian matrix applies to signed graphs
� The Laplacian spectrum denotes graph conflict
� The signed Laplacian arises in several ways:� For graph drawing, the Laplacian implements antipodal proximity� For clustering, the Laplacian implements signed cuts� As an interpretation of negation as inversion of electrical
potential
Thank You
Kunegis et al. Spectral Analysis of Signed Graphs 23
References
P. Hage, F. Harary. Structural models in anthropology, Cambridge University Press, 1983.
F. Harary. On the notion of balance of a signed graph, Michigan Math. J., 2:143–146, 1953.
J. Kunegis, A. Lommatzsch, C. Bauckhage, The Slashdot Zoo: Mining a social network with negative edges, Proc. Int. World Wide Web Conf., pages 741–750, 2009.
J. Leskovec, Daniel Huttenlocher, Jon Kleinberg, Predicting positive and negative links in online social networks, Proc. Int. World Wide Web Conf., 2010.
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Appendix – Balance vs Volume
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Appendix – Scalability
• Evaluation results in function of reduced rank k
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Appendix -- Balance, Conflict and the Graph Spectrum
Look at triads of users (Harary 1953):
• In balanced triangles, the multiplication rule holds
• If it doesn't, there is conflict
Balance:
Conflict:
Kunegis et al. Spectral Analysis of Signed Graphs 27
The Signed Clustering Coefficient
• How many triangles are balanced ?
Cs = (#balanced − #unbalanced) / #possible
• This measure is local, not global (Kunegis 2009)
± uv ?
u v
Kunegis et al. Spectral Analysis of Signed Graphs 28
Introduction: Networks
• Many web sites allow you to have friends :
Example: Facebook