Listing Maximal k-Plexes in Large Real-World Graphs

TL;DR

This paper introduces an efficient algorithm for listing all maximal k-plexes in large real-world graphs, significantly improving over previous methods and enabling better community detection in network analysis.

Contribution

The paper presents the ListPlex algorithm with a novel time complexity bound and practical enhancements for fast enumeration of maximal k-plexes in large graphs.

Findings

Outperforms state-of-the-art solutions by up to orders of magnitude.

Provides a theoretical bound of O^*(γ^D) for listing maximal k-plexes.

Demonstrates effectiveness on large real-world graphs.

Abstract

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(\gamma^D)$ time for each constant $k$, where $\gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.