Magnetic eigenmaps for community detection in directed networks

TL;DR

This paper introduces magnetic eigenmaps, a spectral clustering method using the magnetic Laplacian, to detect diverse community structures in directed networks, including cycles and flow-based roles.

Contribution

It presents a novel spectral clustering approach leveraging the magnetic Laplacian for identifying complex communities in directed networks.

Findings

Successfully detects dense communities with cycles

Reveals role communities in flow networks

Provides a multi-scale community analysis method

Abstract

Communities in directed networks have often been characterized as regions with a high density of links, or as sets of nodes with certain patterns of connection. Our approach for community detection combines the optimization of a quality function and a spectral clustering of a deformation of the combinatorial Laplacian, the so-called magnetic Laplacian. The eigenfunctions of the magnetic Laplacian, that we call magnetic eigenmaps, incorporate structural information. Hence, using the magnetic eigenmaps, dense communities including directed cycles can be revealed as well as "role" communities in networks with a running flow, usually discovered thanks to mixture models. Furthermore, in the spirit of the Markov stability method, an approach for studying communities at different energy levels in the network is put forward, based on a quantum mechanical system at finite temperature.