# Markov chain approach to anomalous diffusion on Newman-Watts networks

## TL;DR

This paper uses a Markov chain framework to analyze anomalous diffusion of random walkers on Newman-Watts networks, providing precise conditions for sub- or super-diffusive regimes and a new scaling approach for large networks.

## Contribution

It introduces a Markov chain formalism for analyzing anomalous diffusion on Newman-Watts networks, enabling single-sample analysis and a new scaling ansatz for large network behavior.

## Key findings

- Exact MSD expressions for simple cycle graphs.
- Distinction between genuine anomalous regimes and transient effects.
- New scaling law for walker dynamics in large networks.

## Abstract

A Markov chain (MC) formalism is used to investigate the mean-square displacement (MSD) of a random walker on Newman-Watts (NW) 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 $N\leq1500$ nodes, our very precise results justify a new and more comprehensive scaling ansatz for walker dynamics, from which the behavior for very large networks can be derived.