Random walk and trapping processes on scale-free networks

TL;DR

This paper studies the behavior of random walks on scale-free networks, revealing unique coverage and trapping dynamics through extensive simulations, and compares numerical results with theoretical approximations.

Contribution

It provides new insights into random walk and trapping processes on scale-free networks, including numerical analysis and comparison with theoretical models.

Findings

Random walkers stay near their origin but cover large network areas.

Survival probability follows exponential decay for 2<γ<3.

Different behaviors observed for γ>3 requiring advanced modeling.

Abstract

In this work we investigate the dynamics of random walk processes on scale-free networks in a short to moderate time scale. We perform extensive simulations for the calculation of the mean squared displacement, the network coverage and the survival probability on a network with a concentration $c$ of static traps. We show that the random walkers remain close to their origin, but cover a large part of the network at the same time. This behavior is markedly different than usual random walk processes in the literature. For the trapping problem we numerically compute $\Phi(n,c)$, the survival probability of mobile species at time $n$, as a function of the concentration of trap nodes, $c$. Comparison of our results to the Rosenstock approximation indicate that this is an adequate description for networks with $2<\gamma<3$ and yield an exponential decay. For $\gamma>3$ the behavior is more complicated and one needs to employ a truncated cumulant expansion.