Encoding dynamics for multiscale community detection: Markov time sweeping for the Map equation

TL;DR

This paper enhances multiscale community detection by integrating Markov dynamics into the Map equation, enabling the identification of communities at various scales and overcoming limitations of the original one-step coding scheme.

Contribution

It introduces a Markov time sweeping method that incorporates multistep transition dynamics into the Map equation for improved multiscale community detection.

Findings

The original Map equation neglects internal community structure and has a field-of-view limit.

The proposed method reveals multiscale community structures beyond the original detection scale.

Small compression gap indicates relevant community partitions at different Markov times.

Abstract

The detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules. This notion has been recently exploited by the Map equation formalism (M. Rosvall and C.T. Bergstrom, PNAS, 105(4), pp.1118--1123, 2008) through an information-theoretic description of the process of coding inter- and intra-community transitions of a random walker in the network at stationarity. However, a thorough study of the relationship between the full Markov dynamics and the coding mechanism is still lacking. We show here that the original Map coding scheme, which is both block-averaged and one-step, neglects the internal structure of the communities and introduces an upper scale, the `field-of-view' limit, in the communities it can detect. As a consequence, Map is well tuned to detect clique-like communities but can lead to undesirable overpartitioning when communities are far from clique-like. We show that a signature of this behavior is a large compression gap: the Map description length is far from its ideal limit. To address this issue, we propose a simple dynamic approach that introduces time explicitly into the Map coding through the analysis of the weighted adjacency matrix of the time-dependent multistep transition matrix of the Markov process. The resulting Markov time sweeping induces a dynamical zooming across scales that can reveal (potentially multiscale) community structure above the field-of-view limit, with the relevant partitions indicated by a small compression gap.