Topological strata of weighted complex networks

TL;DR

This paper introduces a novel topological method using persistent homology to identify non-local structures in weighted complex networks, revealing new classifications beyond traditional local measures.

Contribution

It presents a new approach based on algebraic topology to detect mesoscopic structures in networks, bridging network theory and topological data analysis.

Findings

Identifies two classes of weighted networks based on topological features.

Reveals non-local structures invisible to existing network analysis methods.

Provides a new classification scheme for complex networks.

Abstract

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.