Topological Strata of Weighted Complex Networks

PLoS One. 2013 Jun 21;8(6):e66506. doi: 10.1371/journal.pone.0066506. Print 2013.

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.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Humans
  • Models, Theoretical*
  • Social Support*

Grants and funding

GP is supported by the TOPDRIM project funded by the Future and Emerging Technologies program of the European Commission under Contract IST-318121. ID and MS are partly supported by Project Lagrange Ph.D. Grant. FV is partially supported by PRIN 2009 “Spazi di Moduli e Teoria di Lie”. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.