Approximating temporal modularity on graphs of small underlying treewidth

TL;DR

This paper presents an efficient method to approximate temporal modularity in graphs with small treewidth, extending static graph algorithms to dynamic networks with evolving connections.

Contribution

It introduces a novel approximation algorithm for temporal modularity applicable to graphs with small treewidth, overcoming challenges posed by the temporal aspect.

Findings

Efficient multiplicative approximation for temporal modularity in small treewidth graphs.

Extension of static modularity algorithms to dynamic temporal graphs.

Addresses technical challenges of temporal graph structures.

Abstract

Modularity is a very widely used measure of the level of clustering or community structure in networks. Here we consider a recent generalisation of the definition of modularity to temporal graphs, whose edge-sets change over discrete timesteps; such graphs offer a more realistic model of many real-world networks in which connections between entities (for example, between individuals in a social network) evolve over time. Computing modularity is notoriously difficult: it is NP-hard even to approximate in general, and only admits efficient exact algorithms in very restricted special cases. Our main result is that a multiplicative approximation to temporal modularity can be computed efficiently when the underlying graph has small treewidth. This generalises a similar approximation algorithm for the static case, but requires some substantially new ideas to overcome technical challenges associated with the temporal nature of the problem.