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.: A A A A A A

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

3. eigenvaules of random graphs/discrete matrices - Let’s arrange the eigenvalues of matrices so that Much is known about the large eigenvalues of random graphs, e.g. G(n,½) Wigner semi-circle law Füredi-Komlos more recently, the small (magnitude) eigenvalues of random non-symmetric discrete matrices Rudelson 06, Tao-Vu 06, Rudelson-Vershynin 07 (Littlewood-Offord estimates) T RACE M ETHOD Litvak-Pajor-Rudelson-Tomczak-Jaegermann 05 for singular values of rectangular matrices … but significantly less is understood about the eigenvectors.

4. spectral analysis 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)] Heuristics for random instances of NP-hard problems, e.g. - Refuting random 3-SAT above the threshold - Planted cliques, bisections, assignments, colorings

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

6. 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). Graph with positive and non-positive nodes marked. So this graph/eigenvector pair has 6 domains. Our question: What is the nodal domain structure of the eigenvectors of G(n,p)? If we scale the (non-first) eigenvectors of G(n,½) so they lie on S n-1, do they behave like random vectors on the sphere? Observation: If we choose a random vector on S n-1 and a random graph, then almost surely the number of domains is precisely 2.

7. nodal domains - If f k is the k th eigenvector of G, then a discrete version [Davies-Leydold-Stadler] of Courant’s nodal domain theorem (from Riemannian geometry) says that f k has at most k nodal domains. observations: - 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 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: probability of exceptional vertex number of nodes 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)

9. nodal domains theorem: main lemma (2-norm can’t vanish on large subsets): Almost surely, for every (non-first) eigenvector f of 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.) follows from… x Almost surely, every eigenvector of G(n,p) has 2 primary nodal domains, along with (possibly) O( 1 /p) exceptional vertices.

10. the main lemma and LPRT main lemma: Almost surely, for every (non-first) eigenvector f of 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 VnSVnS

11. the main lemma and LPRT main lemma: Almost surely, for every (non-first) eigenvector f of 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 VnSVnS B is i.i.d. The above inequality yields but this almost surely impossible (even taking a union bound over all S’s)

12. lower bounding singular values Want to show that for a ( 1 +  ) n £ n random sign matrix B, Want to argue that is often large for i.i.d. signs {  1, …,  n } The vectors and have very different behaviors. 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. only the beginning… - Tightening nodal domain structure (e.g. no exceptional vertices), e.g. prove: - We’re missing something big (as experiments show) The case of G n,d : E.g. is the adjacency matrix of a random 3-regular graph almost surely non-singular? d=3d=4 d=5