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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.

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- 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)
- 3. 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
- 4. Kunegis et al. Spectral Analysis of Signed Graphs 4 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
- 5. Kunegis et al. Spectral Analysis of Signed Graphs 5 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
- 6. 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 v2 v3 −v1
- 7. Kunegis et al. Spectral Analysis of Signed Graphs 7 Example: Synthetic Graph Unsigned Graph Drawing → Signed Graph Drawing
- 8. Kunegis et al. Spectral Analysis of Signed Graphs 8 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)
- 9. Kunegis et al. Spectral Analysis of Signed Graphs 9 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
- 10. Kunegis et al. Spectral Analysis of Signed Graphs 10 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
- 11. Kunegis et al. Spectral Analysis of Signed Graphs 11 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
- 12. Kunegis et al. Spectral Analysis of Signed Graphs 12 Finding Communities The Laplacian matrix finds communities: • Communities are connected by many positive edges • Community are separated by many negative edges
- 13. Kunegis et al. Spectral Analysis of Signed Graphs 13 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
- 14. Kunegis et al. Spectral Analysis of Signed Graphs 14 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
- 15. Kunegis et al. Spectral Analysis of Signed Graphs 15 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
- 16. Kunegis et al. Spectral Analysis of Signed Graphs 16 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?
- 17. Kunegis et al. Spectral Analysis of Signed Graphs 17 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 −
- 18. Kunegis et al. Spectral Analysis of Signed Graphs 18 • 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)
- 19. Kunegis et al. Spectral Analysis of Signed Graphs 19 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
- 20. Kunegis et al. Spectral Analysis of Signed Graphs 20 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
- 21. 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
- 22. Thank You
- 23. 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.
- 24. Kunegis et al. Spectral Analysis of Signed Graphs 24 Appendix – Balance vs Volume
- 25. Kunegis et al. Spectral Analysis of Signed Graphs 25 Appendix – Scalability • Evaluation results in function of reduced rank k
- 26. Kunegis et al. Spectral Analysis of Signed Graphs 26 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:
- 27. 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
- 28. Kunegis et al. Spectral Analysis of Signed Graphs 28 Introduction: Networks • Many web sites allow you to have friends: Example: Facebook

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