Networks and Cycles: A Persistent Homology Approach to Complex Networks

Abstract

Persistent homology is an emerging tool to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems (such as brain dynamics, molecular folding, distributed sensing).

Its central device, the filtration, embodies this by casting the analysis of the system in terms of long-lived (persistent) topological properties under the change of a scale parameter.

In the classical case of data clouds in high-dimensional metric spaces, such filtration is uniquely defined by the metric structure of the point space. On networks instead, multiple ways exists to associate a filtration. Far from being a limit, this allows to tailor the construction to the specific analysis, providing multiple perspectives on the same system.

In this work, we introduce and discuss three kinds of network filtrations, based respectively on the intrinsic network metric structure, the hierarchical structure of its cliques and—for weighted networks—the topological properties of the link weights. We show that persistent homology is robust against different choices of network metrics. Moreover, the clique complex on its own turns out to contain little information content about the underlying network. For weighted networks we propose a filtration method based on a progressive thresholding on the link weights, showing that it uncovers a richer structure than the metrical and clique complex approaches.