Self-avoiding walks on scale-free networks

TL;DR

This paper investigates the properties of self-avoiding walks on scale-free networks, revealing how network size and degree distribution influence walk length and attrition, with implications for network exploration.

Contribution

It provides analytical and simulation insights into the behavior of self-avoiding walks on scale-free networks, including growth rates and attrition effects.

Findings

Average number of SAWs grows exponentially with walk length

Maximum walk length scales as a power of network size

Attrition of paths increases with network loops and degree distribution

Abstract

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are expected to be more suitable than unrestricted random walks to explore various kinds of real-life networks. Here we study long-range properties of random SAWs on scale-free networks, characterized by a degree distribution $P(k) \sim k^{-\gamma}$. In the limit of large networks (system size $N \to \infty$), the average number $s_n$ of SAWs starting from a generic site increases as $\mu^n$, with $\mu = < k^2 > / < k > - 1$. For finite $N$, $s_n$ is reduced due to the presence of loops in the network, which causes the emergence of attrition of the paths. For kinetic growth walks, the average maximum length, $< L >$, increases as a power of the system size: $< L > \sim N^{\alpha}$, with an exponent $\alpha$ increasing as the parameter $\gamma$ is raised. We discuss the dependence of $\alpha$ on the minimum allowed degree in the network. A similar power-law dependence is found for the mean self-intersection length of non-reversal random walks. Simulation results support our approximate analytical calculations.