Rate-optimal community detection near the KS threshold via node-robust algorithms

Abstract

We study community detection in the \emph{symmetric $k$-stochastic block model}, where $n$ nodes are evenly partitioned into $k$ clusters with intra- and inter-cluster connection probabilities $p$ and $q$, respectively. Our main result is a polynomial-time algorithm that achieves the minimax-optimal misclassification rate \begin{equation*} \exp \Bigl(-\bigl(1 \pm o(1)\bigr) \tfrac{C}{k}\Bigr), \quad \text{where } C = (\sqrt{pn} - \sqrt{qn})^2, \end{equation*} whenever $C \ge K\,k^2\,\log k$ for some universal constant $K$, matching the Kesten--Stigum (KS) threshold up to a $\log k$ factor. Notably, this rate holds even when an adversary corrupts an $\eta \le \exp\bigl(- (1 \pm o(1)) \tfrac{C}{k}\bigr)$ fraction of the nodes. To the best of our knowledge, the minimax rate was previously only attainable either via computationally inefficient procedures [ZZ15] or via polynomial-time algorithms that require strictly stronger assumptions such as $C \ge K k^3$ [GMZZ17]. In the node-robust setting, the best known algorithm requires the substantially stronger condition $C \ge K k^{102}$ [LM22]. Our results close this gap by providing the first polynomial-time algorithm that achieves the minimax rate near the KS threshold in both settings. Our work has two key technical contributions: (1) we robustify majority voting via the Sum-of-Squares framework, (2) we develop a novel graph bisection algorithm via robust majority voting, which allows us to significantly improve the misclassification rate to $1/\mathrm{poly}(k)$ for the initial estimation near the KS threshold.