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Spectral Graph Theory (Basics)

Spectral Graph Theory (Basics). Charalampos (Babis) Tsourakakis CMU. Outline. Reminders Adjacency matrix Intuition behind eigenvectors: Bipartite Graphs Walks of length k Case Study: Triangles Laplacian Connected Components Intuition: Adjacency vs. Laplacian

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Spectral Graph Theory (Basics)

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  1. Spectral Graph Theory (Basics) Charalampos (Babis) Tsourakakis CMU Charalampos E. Tsourakakis

  2. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  3. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  4. Matrix Representations of G(V,E) Associate a matrix to a graph: • Adjacency matrix • Laplacian • Normalized Laplacian Main focus Charalampos E. Tsourakakis

  5. Matrix as an operator The image of the unit circle (sphere) under any mxn matrix is an ellipse (hyperellipse). e.g., Charalampos E. Tsourakakis

  6. More Reminders • Let M be a symmetric nxn matrix. λ eigenvaluex eigenvector Charalampos E. Tsourakakis

  7. More Reminders 1-Dimensional Invariant Subspaces Diagonal: No rotation y x (λ,u) Ax Ay Charalampos E. Tsourakakis

  8. Keep in mind! • For the rest of slides we are talking for square nxn matricesand unless noticed symmetric ones, i.e, Charalampos E. Tsourakakis

  9. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Cheeger Inequality and Sparsest Cut: • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  10. Adjacency matrix Undirected 4 1 A= 2 3 Charalampos E. Tsourakakis

  11. Adjacency matrix Undirected Weighted 4 10 1 4 0.3 A= 2 3 2 Charalampos E. Tsourakakis

  12. Adjacency matrix Directed 4 1 ObservationIf G is undirected,A = AT 2 3 Charalampos E. Tsourakakis

  13. Spectral Theorem Theorem [Spectral Theorem] • If M=MT, thenwhere 0 0 Reminder 2: xi i-th principal axis λilength of i-th principal axis Reminder 1: xi,xj orthogonal λi λj xi xj Charalampos E. Tsourakakis

  14. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  15. Bipartite Graphs Any graph with no cycles of odd length is bipartite e.g., all trees are bipartite K3,3 1 4 2 5 Can we check if a graph is bipartitevia its spectrum?Can we get the partition of the verticesin the two sets of nodes? 3 6 Charalampos E. Tsourakakis

  16. Bipartite Graphs Why λ1=-λ2=3?Recall: Ax=λx, (λ,x) eigenvalue-eigenvector Adjacency matrix K3,3 1 4 where 2 5 3 6 Eigenvalues: Λ=[3,-3,0,0,0,0] Charalampos E. Tsourakakis

  17. Bipartite Graphs 1 1 3=3x1 3 2 1 1 1 4 1 4 1 1 1 2 5 5 1 1 1 3 6 6 Repeat same argument for the other nodes Charalampos E. Tsourakakis

  18. Bipartite Graphs 1 -1 -3=(-3)x1 -3 -2 -1 -1 1 4 1 4 1 -1 -1 2 5 5 1 -1 -1 3 6 6 Repeat same argument for the other nodes Charalampos E. Tsourakakis

  19. Bipartite Graphs • Observationu2 gives the partition of the nodes in the two sets S, V-S! S V-S Question: Were we just “lucky”? Answer: No Theorem: λ2=-λ1iff G bipartite. u2 gives the partition. Charalampos E. Tsourakakis

  20. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  21. Walks • A walk of length r in a directed graph:where a node can be used more than once. • Closed walk when: 4 4 1 1 Closed walk of length 3 2-1-3-2 Walk of length 2 2-1-4 2 2 3 3 Charalampos E. Tsourakakis

  22. Walks • Theorem G(V,E) directed graph, adjacency matrix A. The number of walks from node u to node v in G with length r is (Ar)uv • Proof Induction on k. See Doyle-Snell, p.165 (i,i1),..,(ir-1,j) (i, i1),(i1,j) (i,j) Charalampos E. Tsourakakis

  23. Walks 4 1 2 3 4 i=3, j=3 i=2, j=4 4 1 1 2 3 2 3 Charalampos E. Tsourakakis

  24. Walks 4 1 2 3 Always 0,node 4 is a sink 4 1 2 3 Charalampos E. Tsourakakis

  25. Walks • Corollary A adjacency matrix of undirected G(V,E) (no self loops), e edges and t triangles. Then the following hold:a) trace(A) = 0 b) trace(A2) = 2ec) trace(A3) = 6t 1 Computing Ar is a bad idea: High memory requirements,expensive! 1 2 1 2 3 Charalampos E. Tsourakakis

  26. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  27. Why is Triangle Counting important?From the Graph Mining Perspective A C B Other applications include: • Hidden Thematic Structure of the Web • Motif Detection, e.g., biological networks • Web Spam Detection Clustering coefficient Transitivity ratio Social Network Analysis fact: “Friends of friends are friends” Charalampos E. Tsourakakis

  28. Theorem [EigenTriangle] Theorem δ(G) = # triangles in graph G(V,E) = eigenvalues of adjacency matrix A Charalampos E. Tsourakakis

  29. Theorem[EigenTriangleLocal] Theorem δ(i) = #Δs vertex i participates at. = i-th eigenvector = j-th entry of Charalampos E. Tsourakakis

  30. Algorithm’s idea Political Blogs Omit! Keep only 3! 3 Almost symmetric around 0! Charalampos E. Tsourakakis

  31. Results: Edges vs. Speedup Observe the trend Charalampos E. Tsourakakis

  32. #Eigenvalues vs. ϱ 2-3 eigenvalues almost ideal results! Charalampos E. Tsourakakis

  33. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  34. Laplacian 4 1 L= D-A= 2 3 Diagonal matrix, dii=di Charalampos E. Tsourakakis

  35. Weighted Laplacian 4 10 1 4 0.3 2 3 2 Charalampos E. Tsourakakis

  36. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  37. Connected Components • LemmaLet G be a graph with n vertices and c connected components. If L is the Laplacian of G, then rank(L)=n-c. • Proof see p.279, Godsil-Royle Charalampos E. Tsourakakis

  38. Connected Components G(V,E) 1 2 3 L= 4 6 #zeros = #components 7 5 eig(L)= Charalampos E. Tsourakakis

  39. Connected Components G(V,E) 1 2 3 L= 0.01 4 6 #zeros = #components Indicates a “good cut” 7 5 eig(L)= Charalampos E. Tsourakakis

  40. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  41. Adjacency vs. Laplacian Intuition V-S Let x be an indicator vector: S Consider now y=Lx k-th coordinate Charalampos E. Tsourakakis

  42. Adjacency vs. Laplacian Intuition G30,0.5 S Consider now y=Lx k Charalampos E. Tsourakakis

  43. Adjacency vs. Laplacian Intuition G30,0.5 S Consider now y=Lx k Charalampos E. Tsourakakis

  44. Adjacency vs. Laplacian Intuition G30,0.5 S Consider now y=Lx k k Laplacian: connectivity, Adjacency: #paths Charalampos E. Tsourakakis

  45. Outline • Reminders • Adjacency matrix • Intuition behind eigenvectors: Bipartite Graphs • Walks of length k • Case Study: Triangles • Laplacian • Connected Components • Intuition: Adjacency vs. Laplacian • Sparsest Cut and Cheeger Inequality : • Derivation, intuition • Example • Normalized Laplacian Charalampos E. Tsourakakis

  46. Why Sparse Cuts? • Clustering, Community Detection • And more: Telephone Network Design, VLSI layout, Sparse Gaussian Elimination, Parallel Computation cut 4 8 1 5 9 2 3 6 7 Charalampos E. Tsourakakis

  47. Quality of a Cut • Edge expansion/Isoperimetric number φ 4 1 2 3 Charalampos E. Tsourakakis

  48. Quality of a Cut • Edge expansion/Isoperimetric number φ 4 1 and thus 2 3 Charalampos E. Tsourakakis

  49. Why λ2? V-S Characteristic Vector x S Edges across cut Then: Charalampos E. Tsourakakis

  50. Why λ2? S V-S cut 4 8 1 5 9 2 3 6 7 x=[1,1,1,1,0,0,0,0,0]T xTLx=2 Charalampos E. Tsourakakis

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