Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs

TL;DR

This paper investigates nonlinear diffusion in complex networks with regular subgraphs, revealing superdiffusion in highly connected nodes and finite size effects that influence transport properties.

Contribution

It introduces a model of transport through generalized trees with regular subgraphs, analyzing superdiffusion and finite size effects using heat-kernel expansion.

Findings

Superdiffusion observed in highly connected nodes

Finite size effects influence transport within supernodes

Heat-kernel expansion regularizes effects in even-dimensional spaces

Abstract

Transport through generalized trees is considered. Trees contain the simple nodes and supernodes, either well-structured regular subgraphs or those with many triangles. We observe a superdiffusion for the highly connected nodes while it is Brownian for the rest of the nodes. Transport within a supernode is affected by the finite size effects vanishing as $N\to\infty.$ For the even dimensions of space, $d=2,4,6,...$, the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat-kernel expansion.