Kinetic-growth self-avoiding walks on small-world networks

TL;DR

This study investigates kinetically-grown self-avoiding walks on small-world networks, revealing how their maximum length scales with network size and rewiring probability, bridging behaviors between regular lattices and random networks.

Contribution

It introduces a detailed analysis of self-avoiding walks on small-world networks, showing how attrition length scales with system size and rewiring probability, and provides both numerical and probabilistic insights.

Findings

Mean attrition length diverges as system size increases for p=1.

Attrition length scales as N^{1/2} at p=1 in large systems.

Close to p=1, attrition length diverges as (1-p)^{-4}.

Abstract

Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value < L >. For random networks, this mean attrition length < L > scales as a power of the network size, and diverges in the thermodynamic limit (large system size N). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, < L > grows with system size as N^{1/2}, diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1-p)^{-4}. Results of approximate probabilistic calculations agree well with those derived from numerical simulations.