Persistence partitions of real and synthetic networks

Abstract

Determining network structures that are neither local nor global is an area of research that has received considerable attention. The study of these intermediate structures has been primarily concerned with the detection of network communities but also includes the examination of network roles, core and peripheral structure, etc. In an increasingly relevant line of research, persistent homology has also been used to analyze the shape of a network in terms of the network’s cycle structure and its higher-dimensional analogues. In this work, we bring these two perspectives together by introducing a non-parametric partition of a network derived from its persistent homology. This partition, which we call the network’s persistence partition, is defined using the concept of a persistence surface , assigning to each node a measure of its individual persistence relative to its position in the network. We examine the extent to which this partition aligns with standard notions of network roles defined via combinatorial equivalence. We then compare how persistence partitions relate to communities and to the core–periphery structure of a network. Our analysis draws on both real and synthetic networks and demonstrates that persistent homology can reveal distinctive structural features that are not detected by conventional methods.