Semidefinite Programs on Sparse Random Graphs

Denote by A the adjacency matrix of an Erdős-Rényi graph with bounded average degree. We consider the problem of maximizing 〈A−E{A}, X〉 over the set of positive semidefinite matrices X with diagonal entries Xii = 1. We prove that for large (bounded) average degree γ, the value of this semidefinite program (SDP) is –with high probability– 2n √ γ + n o( √ γ) + o(n). Our proof is based on two tools from different research areas. First, we develop a new ‘higherrank’ Grothendieck inequality for symmetric matrices. In the present case, our inequality implies that the value of the above SDP is arbitrarily well approximated by optimizing over rank-k matrices for k large but bounded. Second, we use the interpolation method from spin glass theory to approximate this problem by a second one concerning Wigner random matrices instead of sparse graphs. As an application of our results, we prove new bounds on community detection via SDP that are substantially more accurate than the state of the art. 1. Main results 1.1. Semidefinite programs on sparse random graphs. Let G = (V,E) be a random graph with vertex set V = [n], and let AG ∈ {0, 1}n×n denote its adjacency matrix. Spectral algorithms have proven extremely successful in analyzing the structure of such graphs under various probabilistic models. Interesting tasks include finding clusters, communities, latent representations, and so on [AKS98, McS01, NJW+02, CO06]. The underlying mathematical justification for these applications can be informally summarized as follows (more precise statements are given below): If G is dense enough, then (AG − E{AG}) is ‘much smaller’ than E{AG}. However, it was repeatedly observed that this principle breaks down for random graphs with bounded average degree [FO05, CO10, KMO10, DKMZ11, KMM+13], and that spectral methods consequently fail in this case. In order to focus on the simplest non-trivial instance of this phenomenon, assume that G ∼ G(n, γ/n) is an Erdős-Rényi random graph with edge probability γ/n. Then –letting λmax( · ) denote the largest eigenvalue– we have λmax(EAG) = γ. On the other hand, with high probability [KS03, Vu05] λmax(AG − EAG) = { 2 √ γ (1 + o(1)) if γ (log n)4, √ log n/(log log n)(1 + o(1)) if γ = O(1). (1.1) (The same behavior holds for the second-largest eigenvalue λ2(AG)). In particular, λmax(AG − EAG) λmax(EAG) for bounded average degree γ. This phenomenon is not limited to Erdős-Rényi random graphs, and instead leads to failures of spectral methods for Date: April 22, 2015.