Self-avoiding walks on scale-free networks

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 (SAW's) 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 SAW's on scale-free networks, characterized by a degree distribution P(k) approximately k(-gamma). In the limit of large networks (system size N-->infinity), the average number sn of SAW's starting from a generic site increases as mu(n) , with mu = k2/k-1 . For finite N, sn 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 approximately Nalpha, 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 nonreversal random walks. Simulation results support our approximate analytical calculations.