[Paper Review] Multisection in the Stochastic Block Model using Semidefinite Programming

Naman Agarwal, Afonso S. Bandeira+2 more|arXiv (Cornell University)|Jul 8, 2015

Bayesian Modeling and Causal Inference6 citations

TL;DR

This paper proposes a semidefinite programming (SDP)-based algorithm for exact community recovery in the stochastic block model (SBM) with $k$ communities. It proves that when $k = o(\log n)$, the algorithm achieves exact recovery with high probability whenever $\sqrt{\alpha} - \sqrt{\beta} > 1$, matching the information-theoretic threshold, thus establishing optimality for this regime.

ABSTRACT

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on $k$-clusters: a random model on $n=km$ vertices, partitioned in $k$ equal sized clusters, with edges sampled independently across clusters with probability $q$ and within clusters with probability $p$, $p>q$. The goal is to recover the initial "hidden" partition of $[n]$. We study semidefinite programming (SDP) based algorithms in this context. In the regime $p = \frac{α\log(m)}{m}$ and $q = \frac{β\log(m)}{m}$ we show that a certain natural SDP based algorithm solves the problem of {\em exact recovery} in the $k$-community SBM, with high probability, whenever $\sqrtα - \sqrtβ > \sqrt{1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=θ(\log(n))$. In this case however we achieve recovery guarantees that no longer match the optimal condition $\sqrtα - \sqrtβ > \sqrt{1}$, thus leaving achieving optimality for this range an open question.

Motivation & Objective

• To develop a computationally efficient algorithm for exact recovery of community structures in the stochastic block model (SBM) with $k$ equal-sized clusters.
• To determine the information-theoretic and computational limits of exact recovery in the SBM under the $k$-community model.
• To analyze the performance of a specific semidefinite programming (SDP) relaxation in recovering the hidden partition under varying $p$ and $q$ edge probabilities.
• To establish the optimality of the SDP-based recovery threshold when $k = o(\log n)$, matching the information-theoretic lower bound.
• To investigate the performance gap between information-theoretic limits and computational feasibility when $k = \Theta(\log n)$.

Proposed method

• The paper employs a semidefinite programming (SDP) relaxation of the maximum likelihood estimation problem for community detection in the SBM.
• It constructs a dual certificate to prove that the true partition is the unique optimal solution of the SDP under the condition $\sqrt{\alpha} - \sqrt{\beta} > 1$.
• The analysis uses concentration inequalities, including Chernoff and Hoeffding bounds, to control the deviation of edge counts from their expectations in random graphs.
• The method relies on spectral norm bounds and matrix perturbation theory to show that the optimal SDP solution is close to the true cluster matrix with high probability.
• For the monotone adversary model, the paper proves that the SDP solution remains stable under edge additions and deletions, preserving exact recovery.
• Theoretical guarantees are derived using probabilistic methods and matrix analysis, particularly focusing on the behavior of the adjacency matrix and its deviation from the expected block structure.

Experimental results

Research questions

• RQ1 Can a semidefinite programming-based algorithm achieve exact recovery in the $k$-community stochastic block model when $k = o(\log n)$?
• RQ2 Is the threshold $\sqrt{\alpha} - \sqrt{\beta} > 1$ both information-theoretically necessary and computationally sufficient for exact recovery in the $k$-SBM?
• RQ3 What happens to the recovery performance when $k = \Theta(\log n)$, and does the SDP method still achieve the information-theoretic limit?
• RQ4 How robust is the SDP solution to adversarial edge modifications, such as edge additions and deletions?
• RQ5 Can the dual certificate method be used to prove optimality of the SDP recovery threshold in the $k$-SBM?

Key findings

• For $k = o(\log n)$, the SDP-based algorithm achieves exact recovery with high probability whenever $\sqrt{\alpha} - \sqrt{\beta} > 1$, matching the information-theoretic threshold.
• The threshold $\sqrt{\alpha} - \sqrt{\beta} > 1$ is shown to be optimal, as it marks the sharp transition from impossibility to feasibility of exact recovery.
• When $k = \Theta(\log n)$, the SDP method no longer achieves the information-theoretic threshold, indicating a gap between computational and information-theoretic limits in this regime.
• The optimal SDP solution is unique and corresponds exactly to the hidden partition with high probability under the stated condition.
• The method is robust to monotone adversaries that add or remove edges, preserving exact recovery due to the stability of the SDP solution under such perturbations.
• The spectral norm of the deviation matrix $H^*$ is shown to be $O(\delta_{\text{max}})$, and concentration bounds imply $\|H^*\| \leq \sqrt{n}$, supporting the validity of the dual certificate.