Random Walk Access Times on Partially-Disordered Complex Networks: an Effective Medium Theory

TL;DR

This paper develops an effective medium theory to analyze mean access times for random walks on hybrid disordered networks embedded in lattices, comparing their efficiency to small-world networks.

Contribution

It introduces a novel effective medium theory for arbitrary shortcut distributions in hybrid networks and validates it with numerical simulations.

Findings

Numerical simulations agree well with the theory across all aspects.

Partially disordered networks can have comparable access times to small-world networks.

Small-world networks are most effective in reducing access times uniformly.

Abstract

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are different for steps across lattice bonds from the rates $f$ across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially-disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.