Markov chain approach to anomalous diffusion on Newman–Watts networks

Abstract

A Markov chain (MC) formalism is used to investigate the mean-square displacement (MSD) of a random walker on Newman–Watts networks. It leads to a precise analysis of the conditions for the emergence of anomalous sub- or super-diffusive regimes in such random media. Whereas results provided by most numerical approaches used so far base their results on the computation of a large number of independent runs over many equivalent substrates, the MC framework is applied only once to each equivalent sample. Starting from the simple cycle graph with 2k nearest neighbor connections, for which exact MSD expressions within the MC formalism can be derived, the randomness and complexity of the substrate is easily controlled by the number x of added links. Results for different values of k, x, and the number N of nodes make it possible to distinguish actual anomalous regimes from transient behavior and finite size effects. Albeit the high computing cost restricts the size of our networks to nodes, our very precise results justify a new and more comprehensive scaling ansatz for the walker dynamics, from which the behavior for very large networks can be derived.