Provably and Efficiently Approximating Near-cliques using the Tur\'an Shadow: PEANUTS

TL;DR

This paper introduces an efficient, provable method for counting near-cliques with 1 or 2 missing edges in large graphs, significantly outperforming previous algorithms by adapting the Turán Shadow sampling technique.

Contribution

It presents the first space-efficient, scalable algorithm for near-clique counting using a novel adaptation of the Turán Shadow approach, enabling faster analysis of large graphs.

Findings

Achieves 10x to 100x speedup over existing methods.

Successfully counts near-cliques in graphs with tens of millions of edges.

Provides a provable approximation framework for near-clique enumeration.

Abstract

Clique and near-clique counts are important graph properties with applications in graph generation, graph modeling, graph analytics, community detection among others. They are the archetypal examples of dense subgraphs. While there are several different definitions of near-cliques, most of them share the attribute that they are cliques that are missing a small number of edges. Clique counting is itself considered a challenging problem. Counting near-cliques is significantly harder more so since the search space for near-cliques is orders of magnitude larger than that of cliques. We give a formulation of a near-clique as a clique that is missing a constant number of edges. We exploit the fact that a near-clique contains a smaller clique, and use techniques for clique sampling to count near-cliques. This method allows us to count near-cliques with 1 or 2 missing edges, in graphs with tens of millions of edges. To the best of our knowledge, there was no known efficient method for this problem, and we obtain a 10x - 100x speedup over existing algorithms for counting near-cliques. Our main technique is a space-efficient adaptation of the Tur\'an Shadow sampling approach, recently introduced by Jain and Seshadhri (WWW 2017). This approach constructs a large recursion tree (called the Tur\'an Shadow) that represents cliques in a graph. We design a novel algorithm that builds an estimator for near-cliques, using an online, compact construction of the Tur\'an Shadow.