We study the application of spectral clustering, prediction and visualization methods to graphs with negatively weighted edges. We show that several characteristic matrices of graphs can be extended to graphs with positively and negatively weighted edges, giving signed spectral clustering methods, signed graph kernels and network visualization methods that apply to signed graphs. In particular, we review a signed variant of the graph Laplacian. We derive our results by considering random walks, graph clustering, graph drawing and electrical networks, showing that they all result in the same formalism for handling negatively weighted edges. We illustrate our methods using examples from social networks with negative edges and bipartite rating graphs.

1. Spectral Analysis of Signed Graphs for
Clustering, Prediction and Visualization
Jérôme Kunegis¹, Stephan Schmidt¹, Andreas Lommatzsch¹ & Jürgen Lerner²
¹DAI Lab, Technische Universität Berlin, ²Universität Konstanz, Germany
10th
SIAM International Conference on Data Mining, April 29–May 1, Columbus, Ohio

2. Kunegis et al. Spectral Analysis of Signed Graphs 2
Introduction: Negative Edges
Some websites allow you to have foes:
Example: Slashdot Zoo (Kunegis 2009)

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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

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Outline
Introduction: Signed Graphs
1. Negative Edges and the Laplacian
2. Balance, Conflict and the Graph Spectrum
3. Communities, Cuts and Clustering
4. 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

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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
v2
v3
−v1

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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
xT
Lx = 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
−w
w −

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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

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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

22. Thank You

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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:

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The Signed Clustering Coefficient
• How many triangles are balanced?
Cs
= (#balanced − #unbalanced) / #possible
• This measure is local, not global (Kunegis 2009)
± uv ?
u v

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Introduction: Networks
• Many web sites allow you to have friends:
Example: Facebook