Bounds and algorithms for graph trusses

TL;DR

This paper explores the combinatorial properties of graph $k$-trusses, providing bounds on their edge counts and introducing two improved algorithms for identifying trusses efficiently in large graphs.

Contribution

It offers nearly-tight bounds on $k$-truss edge counts and introduces two novel algorithms, including a theoretical one based on fast matrix multiplication.

Findings

Nearly-tight bounds on $k$-truss edge counts

A simplified, faster truss-finding algorithm

A theoretical algorithm using fast matrix multiplication

Abstract

The $k$-truss, introduced by Cohen (2005), is a graph where every edge is incident to at least $k$ triangles. This is a relaxation of the clique. It has proved to be a useful tool in identifying cohesive subnetworks in a variety of real-world graphs. Despite its simplicity and its utility, the combinatorial and algorithmic aspects of trusses have not been thoroughly explored. We provide nearly-tight bounds on the edge counts of $k$-trusses. We also give two improved algorithms for finding trusses in large-scale graphs. First, we present a simplified and faster algorithm, based on approach discussed in Wang & Cheng (2012). Second, we present a theoretical algorithm based on fast matrix multiplication; this converts a triangle-generation algorithm of Bjorklund et al. (2014) into a dynamic data structure.