# Total variation based community detection using a nonlinear optimization approach

## TL;DR

This paper introduces a novel nonlinear optimization method based on total variation to detect communities in networks, providing an effective alternative to existing modularity maximization techniques.

## Contribution

It proposes a new total variation based formulation for community detection, with a fast first-order algorithm that outperforms current methods.

## Key findings

- The new method achieves better community detection accuracy.
- It compares favorably with state-of-the-art approaches in numerical tests.
- The approach effectively solves the modularity optimization problem.

## Abstract

Maximizing the modularity of a network is a successful tool to identify an important community of nodes. However, this combinatorial optimization problem is known to be NP-complete. Inspired by recent nonlinear modularity eigenvector approaches, we introduce the modularity total variation $TV_Q$ and show that its box-constrained global maximum coincides with the maximum of the original discrete modularity function. Thus we describe a new nonlinear optimization approach to solve the equivalent problem leading to a community detection strategy based on $TV_Q$. The proposed approach relies on the use of a fast first-order method that embeds a tailored active-set strategy. We report extensive numerical comparisons with standard matrix-based approaches and the Generalized RatioDCA approach for nonlinear modularity eigenvectors, showing that our new method compares favourably with state-of-the-art alternatives.