Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks

TL;DR

This paper introduces a dynamic, Markov process-based framework for community detection in complex networks, enabling multi-scale analysis and unifying various existing methods through a time-parametrized stability measure.

Contribution

It presents a systematic dynamical approach to community detection using Markov Stability, linking stochastic dynamics with community structures at multiple scales.

Findings

Markov Stability efficiently uncovers multi-scale community structures.

The framework unifies spectral algorithms and heuristics like modularity and Potts model.

It favors communities with similar centrality, providing a versatile detection method.

Abstract

Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process provides a dynamical sweeping across all scales in the graph, and the time scale is an intrinsic parameter that uncovers communities at different resolutions. This dynamic-based community detection leads to a compound optimization, which favours communities of comparable centrality (as defined by the stationary distribution), and provides a unifying framework for spectral algorithms, as well as different heuristics for community detection, including versions of modularity and Potts model. Our dynamic framework creates a systematic link between different stochastic dynamics and their corresponding notions of optimal communities under distinct (node and edge) centralities. We show that the Markov Stability can be computed efficiently to find multi-scale community structure in large networks.