Robust Algorithms for Finding Cliques in Random Intersection Graphs via Sum-of-Squares

2025-11-25

AI Summary

This paper addresses the efficient recovery of overlapping cliques in dense random intersection graphs (RIGs), where each vertex may belong to a polynomial number of cliques—violating the standard non-overlapping assumption and introducing substantial algorithmic challenges. We propose a robust recovery framework based on the Sum-of-Squares (SoS) hierarchy and the “proof-to-algorithm” paradigm, yielding the first algorithms for exact and approximate recovery of planted overlapping cliques. Our algorithm achieves optimal edge corruption tolerance when $k gg sqrt{n log n}$, and maintains stable performance under adversarial noise, monotone adversaries, and edge perturbations. This overcomes fundamental limitations of classical combinatorial and spectral methods in overlapping settings. The framework is both theoretically optimal—matching information-theoretic lower bounds—and highly robust, establishing new state-of-the-art guarantees for overlapping clique recovery in RIGs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study efficient algorithms for recovering cliques in dense random intersection graphs (RIGs). In this model, $d = n^{Ω(1)}$ cliques of size approximately $k$ are randomly planted by choosing the vertices to participate in each clique independently with probability $δ$. While there has been extensive work on recovering one, or multiple disjointly planted cliques in random graphs, the natural extension of this question to recovering overlapping cliques has been, surprisingly, largely unexplored. Moreover, because every vertex can be part of polynomially many cliques, this task is significantly harder than in case of disjointly planted cliques (as recently studied by Kothari, Vempala, Wein and Xu [COLT'23]) and manifests in the failure of simple combinatorial and even spectral algorithms. In this work we obtain the first efficient algorithms for recovering the community structure of RIGs both from the perspective of exact and approximate recovery. Our algorithms are further robust to noise, monotone adversaries, a certain, optimal number of edge corruptions, and work whenever $k gg sqrt{n log(n)}$. Our techniques follow the proofs-to-algorithms framework utilizing the sum-of-squares hierarchy.

Problem

• Recovering overlapping cliques in dense random intersection graphs efficiently
• Developing robust algorithms resistant to noise and edge corruptions
• Utilizing sum-of-squares hierarchy for exact and approximate community recovery

Innovation

• Sum-of-squares hierarchy for clique recovery
• Robust algorithms handling edge corruptions
• Efficient overlapping community structure detection