Relative, local and global dimension in complex networks

TL;DR

This paper introduces a scale-dependent, relative dimension concept for complex networks using diffusive processes, revealing insights into structural flexibility, communication, and spreading dynamics across various systems.

Contribution

It proposes a novel, scale-dependent relative and local dimension framework for complex networks based on diffusion, applicable to diverse physical and social systems.

Findings

Local dimension correlates with structural flexibility in proteins.

Relative dimension uncovers regions involved in allosteric communication.

Dimension measures predict spreading capability in epidemic models.

Abstract

Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. To take into account locality, finiteness and discreteness, dynamical processes can be used to probe the space geometry and define its dimension. Here we show that each point in space can be assigned a relative dimension with respect to the source of a diffusive process, a concept that provides a scale-dependent definition for local and global dimension also applicable to networks. To showcase its application to physical systems, we demonstrate that the local dimension of structural protein graphs correlates with structural flexibility, and the relative dimension with respect to the active site uncovers regions involved in allosteric communication. In simple models of epidemics on networks, the relative dimension is predictive of the spreading capability of nodes, and identifies scales at which the graph structure is predictive of infectivity. We further apply our dimension measures to neuronal networks, economic trade, social networks, ocean flows, and to the comparison of random graphs.