Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks

TL;DR

This study investigates how random walk traversal times on hybrid scale-free/regular-lattice networks depend on shortcut density and transition rates, revealing universal behaviors and effects of network inhomogeneity.

Contribution

It introduces a numerical analysis of mean access times on hybrid networks, highlighting universal scaling and the impact of shortcut distribution inhomogeneity.

Findings

Universal curve for traversal times at high shortcut speeds and low densities

Crossover behavior related to percolation threshold at higher shortcut densities

Higher inhomogeneity in shortcut distribution increases traversal times

Abstract

We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds ($F$) and across network shortcuts ($f$). For fast shortcuts ($f/F\gg 1 $) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.