EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY
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1 EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY Franklin H. J. Kenter Rice University ( United States Naval Academy 206) orange@rice.edu Connections in Discrete Mathematics, A Celebration of Ron Graham Simon Fraser University; Burnaby, BC June 6, 205
2 WHAT IS SPECTRAL GRAPH THEORY? Answer: Application of Linear Algebra, in particular eigenvalues and eigenvectors, to the study of graphs and networks C A Eigenvalues: 2 + p 7, 2 p 7,, 0 These aren't the juggling patterns you're looking for.
3 WHAT ARE EIGENVALUES GOOD FOR? Community Detection Cheeger s Inequality Principle Component Analysis Bipartitioning Dense Subgraphs Graph Coloring Mixing Times / Expander Mixing Lemma
4 TYPES OF MATRICES C A A Adjacency Matrix Eigenvalues: max apple 2 apple...apple min 0 0 p6 p6 0 p p 6 p6 p p6 p6 0 C A L = I D /2 AD /2 Normalized Laplacian Matrix Eigenvalues: 0 = 0 apple apple... n
5 BASIC FACTS OF THE LAPLACIAN L = I D /2 AD /2 0 = 0 with eigenvector D /2 n apple 2 with equality if G has a bipartite component. = 0 if and only if G is disconnected. k = 0 if and only if G has at least k connected components =0.0307
6 CHEEGER CONSTANT h(s) := E(S, S) min(vol S, vol S) Boundary Area h G = min h(s) S V (G) NP-Hard to compute
7 CHEEGER S INEQUALITY Theorem (Alon-Milman 985) h 2 G 2 apple apple 2h G C A
8 DRAWBACK TO CHEEGER S INEQUALITY Proposition Under mild conditions, h G apple ( + o()) 2 Just flip a coin for each vertex
9 LINEAR CHEEGER INEQUALITY Theorem (K. 205+) Let be the second minimum eigenvalue of the normalized Laplacian with unit eigenvector D /2 v. Then, under mild conditions, 2 2 apple h G apple 2 2kvk 2 vol G ( + o()). Cheeger constant is a measure of how much better than random you can get!
10 PROOF IDEA: Just flip a weighted coin for each vertex where the weight for vertex i is given by 2 + v i ( ") 2kvk
11 PROOF IDEA (CON D): Lemma (Random Quadratic Forms) Given a real symmetric matrix A. For a random vector x, E[x Ax] =µ Aµ +Tr( A) Expectation of Quadratic Form Variance-Covariance Matrix Entry-wise expectation of x If A has diagonal zero entries, and x has independent components...
12 A NICE CONSEQUENCE Theorem (K. 205+) Let be the second minimum eigenvalue of the normalized Laplacian with unit eigenvector D /2 v. Then, under mild conditions, 2 2 apple h G apple 2 2kvk 2 vol G ( + o()).
13 A NICE CONSEQUENCE Corollary If i =+o() for i 6= 0(i.e.,G is an expander), then under mild conditions: h G = 2 ( + o())
14 K-FOLD CHEEGER CONSTANT Cheeger Constant is for 2 parts. What about minimizing edges between 3, 4, or k parts? Higher Order Cheeger Constant (Worst Case Approach): ĥ (k) G = min max P=(S,S 2,...S k ) i h(s i )
15 K-FOLD CHEEGER INEQUALITY Theorem (Lee-Gharan-Trevisan 202) 2 k apple ĥ(k+) G apple O(k 2 ) p 2 k Similar work by Louis-Raghavendra-Tetali-Vempala (20) Can this be improved to a linear factor as previously?
16 K-FOLD CHEEGER CONSTANT: ANOTHER APPROACH Higher Order Cheeger Constant (Average Case Approach): h (k) G (S) = k X i6=j e(s i,s j ) min{vol S i, vol S j }. Average over all parts over Cheeger ratio among those parts. Low Average Case Cheeger Constant High Worst Case Cheeger Constant
17 LINEAR K-FOLD CHEEGER INEQUALITY Theorem (K.-Radcliffe 205+) Let = P k i= kx ik, and let = k Then, under mild conditions, P k i= ( i) 2 2 apple h(k) G apple apple 2 4k (k ) 4vol(G) 2 ( + o()). Also determines how much better than random you can get. Uses similar, but more detailed, approach as before.
18 WHAT ELSE CAN YOU DO WITH EIGENVECTOR NORMS? Densest Subgraph Problem Expander Mixing Lemma Graph Coloring
19 DENSE SUBGRAPH PROBLEM Densest Subgraph Problem M := max S V,S6=; E(S, S) S Goldburg (984): polynomial time algorithm to find the densest subgraph. NP-complete when S has additional restrictions
20 DENSE SUBGRAPH PROBLEM Densest Subgraph Problem M := max S V,S6=; E(S, S) S Theorem (K. 205+) Corresponding Unit Eigenvector max kvk kvk apple M apple max Maximum Density Maximum Eigenvalue of A
21 DENSE SUBGRAPH PROBLEM Densest Subgraph Problem M := max S V,S6=; E(S, S) S Theorem (K. 205+) kvk kvk max M kvk 2 2
22 EXPANDER MIXING LEMMA AND ITS CONVERSE Theorem (Bilu-Linial 2004) Let G be a d-regular undirected graph. Define that for all sets of vertices S, T : is the least constant such E(S, T ) then 2 := max[ 2, min ] obeys apple 2 apple O d S T n apple p S T + log d Expanded by Bollobás-Nikiforov (2004) Butler (2007) and Chung-K. (204) for directed graphs. Chung (204) for graph limits. and more
23 EXPANDER MIXING LEMMA WITH EIGENVECTOR NORMS Theorem (K. 5+) Let G be a d-regular undirected graph. Define such that for all sets of vertices S, T : the be the least constant E(S, T ) d S T n apple p S T then 2 := max[ 2, min ] with unit eigenvector v obeys 2 kvk kvk apple apple 2 No logarithmic factor!
24 MAXCUT / ALMOST 2- COLORINGS Maximum Cut Problem B = max S Ṡ S2 =V 2E(S,S 2 ) E(S,S ) E(S 2,S 2 ) n B apple min Minimum Eigenvalue Trevisan (2008), Bauer-Hua-Jöst (202): A converse applies. If the minimum eigenvalue is large, then a large subgraph has a large maximum cut.
25 MAXCUT / ALMOST 2- COLORINGS Maximum Cut Problem B = max S Ṡ S2 =V 2E(S,S 2 ) E(S,S ) E(S 2,S 2 ) n Theorem (K. 205+) min nkvk 2 apple B apple min Corresponding Unit Eigenvector Minimum Eigenvalue of A
26 MORAL Many spectral bounds are tight when corresponding eigenvectors are well-behaved. It s okay not to be norm al.
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