"A Method Based on Total Variation for Network Modularity Optimization Using the MBO Scheme"

One of my papers just came out in final form. Here are the details.

Title: A Method Based on Total Variation for Network Modularity Optimization Using the MBO Scheme

Authors: Huiyi Hu, Thomas Laurent, Mason A. Porter, and Andrea L. Bertozzi

Abstract: The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called "communities." One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an l_2 balance term. By employing numerical techniques from image processing and l_1 compressive sensing --- such as convex splitting and the Merriman-Bence-Osher (MBO) scheme --- we develop a variational algorithm for the minimization problem. We present our computational results using both synthetic benchmark networks and real data.


The really cool thing about this paper is what it does for "technology translation". We reformulate the modularity quality function in a way that relates it to compressive-sensing-like problems. One can thereby adapt methods from the latter to use for modularity optimization. One of the best compliments that we have received about this paper thus far occurred right after posting it on the arXiv: Somebody from the image-processing community told us that our paper was written in a language such that he could finally understand what people were doing with graphs. Given the goals of the paper, that was exactly what I wanted to hear!