[Paper Review] Semidefinite Programs on Sparse Random Graphs and their Application to Community Detection

Andrea Montanari, Subhabrata Sen|arXiv (Cornell University)|Apr 22, 2015

Complex Network Analysis Techniques6 citations

TL;DR

This paper analyzes semidefinite programs (SDPs) on sparse random graphs, showing that for Erdős-Rényi and random regular graphs with bounded average degree $d$, the SDP value concentrates at $2n\sqrt{d} + o(n\sqrt{d})$. It establishes that SDP achieves the information-theoretically optimal threshold for community detection in the sparse stochastic block model, detecting communities when $(a-b)^2/(4d) > 1 + o_d(1)$, nearly matching the fundamental limit of $(a-b)^2/(4d) > 1$. The analysis combines a higher-rank Grothendieck inequality, interpolation methods from statistical physics, and eigenvector analysis of deformed random matrices.

ABSTRACT

Denote by $A$ the adjacency matrix of an Erdos-Renyi graph with bounded average degree. We consider the problem of maximizing $\langle A-E\{A\},X angle$ over the set of positive semidefinite matrices $X$ with diagonal entries $X_{ii}=1$. We prove that for large (bounded) average degree $d$, the value of this semidefinite program (SDP) is --with high probability-- $2n\sqrt{d} + n\, o(\sqrt{d})+o(n)$. For a random regular graph of degree $d$, we prove that the SDP value is $2n\sqrt{d-1}+o(n)$, matching a spectral upper bound. Informally, Erdos-Renyi graphs appear to behave similarly to random regular graphs for semidefinite programming. We next consider the sparse, two-groups, symmetric community detection problem (also known as planted partition). We establish that SDP achieves the information-theoretically optimal detection threshold for large (bounded) degree. Namely, under this model, the vertex set is partitioned into subsets of size $n/2$, with edge probability $a/n$ (within group) and $b/n$ (across). We prove that SDP detects the partition with high probability provided $(a-b)^2/(4d)> 1+o_{d}(1)$, with $d= (a+b)/2$. By comparison, the information theoretic threshold for detecting the hidden partition is $(a-b)^2/(4d)> 1$: SDP is nearly optimal for large bounded average degree. Our proof is based on tools from different research areas: $(i)$ A new `higher-rank' Grothendieck inequality for symmetric matrices; $(ii)$ An interpolation method inspired from statistical physics; $(iii)$ An analysis of the eigenvectors of deformed Gaussian random matrices.

Motivation & Objective

• To understand the behavior of semidefinite programs (SDPs) on sparse random graphs, particularly Erdős-Rényi and random regular graphs with bounded average degree.
• To establish the exact asymptotic value of the SDP maximizing $\langle \mathbf{A} - \mathbb{E}[\mathbf{A}], \mathbf{X} \rangle $ over rank-one positive semidefinite matrices with unit diagonal entries.
• To determine whether SDP can achieve the information-theoretic detection threshold in the sparse, two-community stochastic block model.
• To develop new analytical tools—specifically a higher-rank Grothendieck inequality and an interpolation method—enabling sharp analysis of SDP on sparse graphs.

Proposed method

• Derive the asymptotic value of the SDP on Erdős-Rényi graphs $G(n, d/n)$ as $2n\sqrt{d} + n\,o(\sqrt{d}) + o(n)$ with high probability for large bounded $d$.
• Prove that for random regular graphs $G^{{\rm reg}}(n,d)$, the SDP value is $2n\sqrt{d-1} + o(n)$, matching the spectral upper bound.
• Introduce a higher-rank Grothendieck inequality for symmetric matrices to bound the error in rank-constrained SDP approximations.
• Apply an interpolation method inspired by statistical physics to smoothly deform the graph model and control the SDP value under perturbations.
• Analyze the eigenvectors of deformed Gaussian random matrices to understand the structure of the optimal solution in the community detection setting.
• Use Lindeberg’s method to replace the centered adjacency matrix of the sparse graph with a Gaussian matrix model, enabling precise asymptotic analysis.

Experimental results

Research questions

• RQ1 What is the asymptotic value of the semidefinite program $\max \langle \mathbf{A} - \mathbb{E}[\mathbf{A}], \mathbf{X} \rangle $ over $\mathbf{X} \succeq 0$, $\operatorname{diag}(\mathbf{X}) = \mathbf{1}$, on sparse Erdős-Rényi graphs?
• RQ2 How does the SDP value on random regular graphs compare to that on Erdős-Rényi graphs, and does it match the spectral upper bound?
• RQ3 Can semidefinite programming detect communities in the sparse stochastic block model when the signal-to-noise ratio is near the information-theoretic threshold?
• RQ4 What new mathematical tools are required to analyze SDPs on sparse graphs where spectral methods fail due to degree fluctuations?
• RQ5 To what extent does the SDP value on sparse graphs depend on the graph's degree distribution, and can this be captured via a Gaussian model?

Key findings

• For Erdős-Rényi graphs with average degree $d$, the SDP value is $2n\sqrt{d} + n\,o(\sqrt{d}) + o(n)$ with high probability as $n \to \infty$.
• For random regular graphs of degree $d$, the SDP value is $2n\sqrt{d-1} + o(n)$, matching the spectral upper bound of $2\sqrt{d-1}$.
• The SDP achieves the information-theoretically optimal detection threshold for the sparse two-community stochastic block model: detection is possible when $(a-b)^2/(4d) > 1 + o_d(1)$.
• The threshold $ (a-b)^2/(4d) > 1 $ is the fundamental limit for community detection; thus, SDP is nearly optimal for large bounded $d$.
• The analysis relies on a new higher-rank Grothendieck inequality, which bounds the error in low-rank SDP approximations.
• The proof uses an interpolation method and Lindeberg’s approach to replace the adjacency matrix with a Gaussian matrix model, enabling precise asymptotic control.