Optimizing Edge Sets in Networks to Produce Ground Truth Communities Based on Modularity

TL;DR

This paper introduces new methods for modifying networks through edge addition or removal to control and understand community structures based on modularity, with applications to social and illicit networks.

Contribution

It formulates new optimization problems for edge modifications to influence community detection and provides algorithms and heuristics to solve them.

Findings

Edge addition can enforce desired community partitions.

Sparsest network representations can preserve community structure.

Modularity exhibits counter-intuitive behaviors complicating heuristic development.

Abstract

We consider two new problems regarding the impact of edge addition or removal on the modularity of partitions (or community structures) in a network. The first problem seeks to add edges to enforce that a desired partition is the partition that maximizes modularity. The second problem seeks to find the sparsest representation of a network that has the same partition with maximum modularity as the original network. We present integer programming formulations, a row generation algorithm, and heuristic algorithms to solve these problems. Further, we demonstrate a counter-intuitive behavior of modularity that makes the development of heuristics for general networks difficult. We then present results on a selection of social and illicit networks from the literature.