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

2025-11-20

AI Summary

This paper studies community detection under the symmetric $k$-random block model with adversarial node contamination. Addressing the low signal-to-noise regime near the Kesten–Stigum (KS) threshold, we propose the first polynomial-time algorithm achieving minimax-optimal misclassification rate $expig(-(1pm o(1))C/kig)$ under the mild sample complexity condition $C geq K k^2 log k$. Our method integrates sum-of-squares programming with robust majority voting and introduces a novel graph bisection subroutine, significantly enhancing tolerance to adversarial corruption—up to $expig(-(1pm o(1))C/kig)$ fraction of contaminated nodes. In contrast to prior work, our algorithm breaks through both the strong-signal assumption and high computational complexity barriers, achieving, for the first time near the KS threshold, simultaneous statistical optimality, polynomial-time solvability, and strong robustness against adversarial perturbations.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 egin{equation*} exp Bigl(-igl(1 pm o(1)igr) frac{C}{k}Bigr), quad ext{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 $ηle expigl(- (1 pm o(1)) frac{C}{k}igr)$ 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.

Problem

• Achieving minimax-optimal community detection near Kesten-Stigum threshold
• Developing polynomial-time algorithms robust against adversarial node corruption
• Closing computational gaps between theoretical limits and practical implementations

Innovation

• Robustified majority voting via Sum-of-Squares framework
• Novel graph bisection algorithm using robust voting
• Polynomial-time algorithm achieving minimax rate near KS threshold