Understanding graphs through spectral densities

Transcription

1 Understanding graphs through spectral densities David Bindel Department of Computer Science Cornell University SCAN seminar, 29 Feb 216 SCAN 1 / 1

2 Can One Hear the Shape of a Drum? 2 u = λu on Ω, u = on Ω Assume that for each n the eigenvalue λ n for Ω 1 is equal to the eigenvalue µ n for Ω 2. Question: Are the regions Ω 1 and Ω 2 congruent in the sense of Euclidean geometry? I first heard the problem posed this way some ten years ago from Professor Bochner. Much more recently, when I mentioned it to Professor Bers, he said, almost at once: You mean, if you had perfect pitch could you find the shape of a drum. Mark Kac, American Math Monthly, 1966 SCAN 2 / 1

3 Can One Hear the Shape of a Drum? No in general (Gordon, Webb, Wolpert in 1992) Yes with constraints (Zelditch in 29) SCAN 3 / 1

4 What Do You Hear? A B S Size of bottlenecks (Cheeger inequality) h 2 λ 2 SCAN 4 / 1

5 What Do You Hear? Volume (Weyl law) N(x) lim x x d/2 = (2π) d ω d vol(ω), N(x) = {# eigenvalues x} Can also tell lengths of geodesics for a closed Riemannian manifold. SCAN 5 / 1

6 Can One Hear the Shape of a Graph? From eigenvalues of adjacency, Laplacian, normalized Laplacian? SCAN 6 / 1

7 A Bestiary of Matrices Adjacency matrix: A Laplacian matrix: L = D A Unsigned Laplacian: L = D + A Random walk matrix: P = D 1 A Normalized adjacency: Ā = D 1/2 AD 1/2 Normalized Laplacian: L = I Ā = D 1/2 LD 1/2 Modularity matrix: B = A ddt 2n All have examples of co-spectral graphs... through spectrum uniquely identifies quantum graphs SCAN 7 / 1

8 What Do You Hear? Size of separators (Cheeger inequality) L SCAN 8 / 1

9 What Do You Hear? What information hides in the eigenvalue distribution? 1 Discretizations of Laplacian: something like Weyl s law 2 Sparse random graphs: Wigner semicircular distribution 3 Real networks: less well understood But computing all eigenvalues seems expensive! SCAN 9 / 1

10 Reminder: Spectral Mapping Consider a matrix H, and let f be analytic on the spectrum. Then if H = V ΛV 1, f(h) = V f(λ)v 1. (generalizes to non-diagonalizable case) SCAN 1 / 1

11 Another Perspective: Density of States Spectra define a generalized function (a density): tr(f(h)) = f(λ)µ(λ) dx = N f(λ k ) where f is an analytic test function. Smooth out to get a picture: a spectral histogram or kernel density estimate. j=k SCAN 11 / 1

12 Heat Kernels DoS information equivalent to looking at the heat kernel trace: h(s) = tr(exp( sh)) = L[µ](s) where H is a positive semi-definite operator. H = L = h(s)/n = probability of self-return after time s from uniform start SCAN 12 / 1

13 Power Moments DoS information equivalent to looking at the power moments: tr(h j ). Has a natural interpretation for matrices associated with graphs: A: number of length k cycles. Ā or P : return probability for k-step random walk (times N). L:?? SCAN 13 / 1

14 Chebyshev Moments Ordinary moments are not good for numerics prefer Chebyshev: d j = T j (A) where T j (z) = cos(j cos 1 (z)) is the jth Chebyshev polynomial. Compute via three-term recurrence: T (z) = 1 T 1 (z) = z T k+1 z = 2zT k (z) T k 1 (z) SCAN 14 / 1

15 Exploring Spectral Densities Kernel polynomial method (see Weisse, Reviews of Modern Physics) Think of spectral distribution on [ 1, 1] as a generalized function 1 1 µ(x)f(x) dx = 1 N N f(λ k ) k=1 Write f(x) = j=1 c jt j (x) and µ(x) = j=1 d jφ j (x), where 1 1 φ j(x)t k (x) dx = δ jk Estimate d j = tr(t j (H)) by stochastic methods Truncate series for µ(x) and filter (avoid Gibbs) Much cheaper than computing all eigenvalues! SCAN 15 / 1

16 Stochastic Trace and Diagonal Estimation Z R n with independent entries, mean and variance 1. E[(Z HZ) i ] = j h ij E[Z i Z j ] = h ii Var[(Z HZ) i ] = j h 2 ij. Serves as the basis for stochastic estimation of Trace (Hutchinson, others; review by Toledo and Avron) Diagonal (Bekas, Kokiopoulou, and Saad) Power of diagonal estimation is under-appreciated... SCAN 16 / 1

17 Diagonal Estimation and LDoS Diagonal estimation also useful for local DoS ν k (x); in the symmetric case with H = QΛQ T, have f(x)ν k (x) dx = f(h) kk = e T k Qf(Λ)QT e k ν k (x) = n qkj 2 δ(x λ j) j=1 DoS is sum of local densities of states: µ(x) = n ν k (x) k=1 SCAN 17 / 1

18 LDoS Information Can compute common centrality measures with LDoS Estrada centrality: exp(γa) kk Resolvent centrality: [ (I γā) 1] kk Some motifs associated with localized eigenvectors at specific values: Chief example: Null vectors of Ā supported on leaves. Use LDoS + topology to find motifs? What else? SCAN 18 / 1

19 Phase Retrieval in Graph Reconstruction Reconstruct graph from completely resolved LDoS at all nodes? Assume H = QΛQ T No multiple eigenvalues = know Q and Λ Can we recover signs in Q? Feels a little like phase retrieval... Of course, we usually have noisy LDoS estimates! SCAN 19 / 1

20 Other methods Golub and Meurant: Gauss quadrature for ν k (x) via Lanczos Good: No stochastic estimation error (vs KPM) Bad: Separate Lanczos per node Roder and Silver: Max entropy estimation Good: Better resolution from Chebyshev moment estimates Bad: More expensive computation per node Under investigation: Hybrid approach. SCAN 2 / 1

21 Exploring Spectral Densities (with David Gleich) Consider spectrum of normalized Laplacian (random walk matrix) Approximate via KPM and compare to full eigencomputation Things we know Eigenvalues in [ 1, 1]; nonsymmetric in general Stability: change d edges, have λ j d ˆλ j λ j+d kth moment = probability of return after k-step random walk Eigenvalue cluster near 1 well-separated clusters Eigenvalue cluster near triangles connected by one node What else can we hear? SCAN 21 / 1

22 Experimental setup Global DoS 1 Chebyshev moments 1 probe vectors (componentwise standard normal) Histogram with 5 bins Local DoS 1 Chebyshev moments 1 probe vectors (componentwise standard normal) Plot smoothed density on [ 1, 1] Spectrally order nodes by density plot Suggestions for better pics are welcome! SCAN 22 / 1

23 Erdos SCAN 23 / 1

24 Erdos (local) SCAN 24 / 1

25 Internet topology SCAN 25 / 1

26 Internet topology (local) SCAN 26 / 1

27 Marvel characters SCAN 27 / 1

28 Marvel characters (local) SCAN 28 / 1

29 Marvel comics SCAN 29 / 1

30 Marvel comics (local) SCAN 3 / 1

31 PGP SCAN 31 / 1

32 PGP (local) SCAN 32 / 1

33 Yeast SCAN 33 / 1

34 Yeast (local) SCAN 34 / 1

35 And a few more... SCAN 35 / 1

36 Enron s (SNAP) SCAN 36 / 1

37 Reuters911 (Pajek) SCAN 37 / 1

38 US power grid (Pajek) SCAN 38 / 1

39 DBLP 21 (LAW) 6 x N = , nnz = 16154, 8 s (1 moments, 1 probes) SCAN 39 / 1

40 Hollywood 29 (LAW) 4 x N = , nnz = , 293 s (1 moments, 1 probes) SCAN 4 / 1

41 x x What Do You Hear? SCAN 41 / 1