Physical networks as network-of-networks

TL;DR

This paper introduces a network-of-networks framework for physical networks, revealing how volume exclusion leads to heterogeneity and degree-volume correlations that influence dynamics, with applications to real systems.

Contribution

It presents a novel network-of-networks model incorporating physical shape and volume exclusion, and analyzes its impact on network structure and dynamics.

Findings

Volume exclusion induces heterogeneity in node volume and degree.

Degree-volume correlations suppress hubs' role in diffusive spreading.

Real systems exhibit properties consistent with the minimal model.

Abstract

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.