eigenvectors of random graphs: nodal domains

1. eigenvectors of random graphs: nodal domains James R. Lee University of Washington Yael Dekel and Nati Linial Hebrew University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

2. preliminaries Random graphs edges present with probability p random d-regular graph Adjacency matrix A (non-zero) function f : V R is an eigenvector of G if there exists an (eigenvalue)  for which Eigenvectors for every x 2 V, where (x) is the set of neighbors of x.

3. - Let’s arrange the eigenvalues of matrices so that eigenvaules of random graphs/discrete matrices Much is known about the large eigenvalues of random graphs, e.g. G(n,½) Wigner semi-circle law Füredi-Komlos TRACE METHOD more recently, the small (magnitude) eigenvalues of random non-symmetric discrete matrices Litvak-Pajor-Rudelson-Tomczak-Jaegermann 05 for singular values of rectangular matrices Rudelson 06, Tao-Vu 06, Rudelson-Vershynin 07 (Littlewood-Offord estimates) … but significantly less is understood about the eigenvectors.

4. In many areas such as machine learning and computer vision, eigenvectors of graphs are the primary tools for tasks like partitioning and clustering. [Shi and Malik (image segmentation); Coifman et. al (PDE, machine learning); Pothen, Simon and Lou (matrix sparsification)] spectral analysis Heuristics for random instances of NP-hard problems, e.g. - Refuting random 3-SAT above the threshold - Planted cliques, bisections, assignments, colorings

5. If we scale the (non-first) eigenvectors of G(n,½) so they lie on Sn-1, do they behave like random vectors on the sphere? are eigenvectors uniform on the sphere? For example, do we (almost surely) have… ? or open problem: X Discrete version of “quantum chaos” (?)

6. If we scale the (non-first) eigenvectors of G(n,½) so they lie on Sn-1, do they behave like random vectors on the sphere? are eigenvectors uniform on the sphere? Nodal domains If f:V R is an eigenvector of a graph G, then f partitions G into maximal connected components on which f has constant sign (say, positive vs. non-positive). So this graph/eigenvector pair has 6 domains. Our question: What is the nodal domain structure of the eigenvectors of G(n,p)? Graph with positive and non-positive nodes marked. Observation: If we choose a random vector on Sn-1 and a random graph, then almost surely the number of domains is precisely 2.

7. observations: nodal domains • If fkis the kth eigenvector of G, then a discrete • version [Davies-Leydold-Stadler] of Courant’s nodal • domain theorem (from Riemannian geometry) says • that fk has at most k nodal domains. • If G has 2N nodal domains, then it has an independent set of size N, • hence N=O(log n)/p. theorem: Almost surely, every eigenvector of G(n,p) has 2 primary nodal domains, along with (possibly) O(1/p) exceptional vertices. Experiments suggest that there are at most 2 nodal domains even for:

8. nodal domains can be a delicate issue: In the combinatorial Laplacian of G(n,½), exceptional vertices can occur (it’s always the vertex of max degree in the largest eigenvalue) probability of exceptional vertex number of nodes theorem: Almost surely, every eigenvector of G(n,p) has 2 primary nodal domains, along with (possibly) O(1/p) exceptional vertices. Experiments suggest that there are at most 2 nodal domains even for:

9. theorem: nodal domains Almost surely, every eigenvector of G(n,p) has 2 primary nodal domains, along with (possibly) O(1/p) exceptional vertices. follows from… main lemma(2-norm can’t vanish on large subsets): Almost surely, for every (non-first) eigenvector fof G(n,p) and every subset of vertices S With |S|¸(1-)n, we have ||f|S||2¸p()where p() !1 as ! 0 and p() > 0 for  < 0.01. (The point is that p() is independent of n.) x

10. main lemma: the main lemma and LPRT Almost surely, for every (non-first) eigenvector fof G(n,p) and every subset of vertices S With |S|¸(1-)n, we have ||f|S||2¸p()where p() !1 as ! 0 and p() > 0 for  < 0.01. (The point is that p() is independent of n.) Consider p = ½ and |S|=0.99n. S z VnS

11. main lemma: the main lemma and LPRT Almost surely, for every (non-first) eigenvector fof G(n,p) and every subset of vertices S With |S|¸(1-)n, we have ||f|S||2¸p()where p() !1 as ! 0 and p() > 0 for  < 0.01. (The point is that p() is independent of n.) Consider p = ½ and |S|=0.99n. S z B is i.i.d. The above inequality yields but this almost surely impossible (even taking a union bound over all S’s) VnS

12. The vectors and have very different behaviors. Want to show that for a (1+) n £ n random sign matrix B, lower bounding singular values Want to argue that is often large for i.i.d. signs {1, …, n} As  0, need a very good understanding of “bad” vectors. For eps>1, this is easy (Payley-Zygmund, Chernoff, union bound over a net) For 0<eps<1, this requires also a quantitative CLT (for the “spread” vectors) [LPRT] For eps=0, requires a deeper understanding of the additive structure of the coordinates Tao-Vu 06 showed that this is related to the additive structure of the coordinates, e.g. whether (rescaled) coordinates lie in arithmetic progression. (See Rudelson-Vershynin for state of the art)

13. - Tightening nodal domain structure (e.g. no exceptional vertices), e.g. prove: only the beginning… - We’re missing something big (as experiments show) The case of Gn,d: E.g. is the adjacency matrix of a random 3-regular graph almost surely non-singular? d=3 d=4 d=5