Scaling Properties of Random Walks on Small-World Networks

TL;DR

This paper investigates the scaling behavior of random walks on one-dimensional small-world networks through simulations and analytical models, revealing three distinct temporal regimes and their effects on walk properties.

Contribution

It introduces a comprehensive scaling framework for random walks on small-world networks, combining numerical and analytical approaches across different time regimes.

Findings

Three distinct temporal regimes identified in walk behavior.

Scaling forms accurately describe average visited sites and displacement.

Finite network size causes saturation of properties at long times.

Abstract

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in $d$-dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.