Self-avoiding walks and connective constants in small-world networks

TL;DR

This paper investigates how self-avoiding walks behave on small-world networks, revealing how the connective constant varies with rewiring probability and providing both numerical and analytical insights into the network's long-distance properties.

Contribution

It introduces a combined numerical and analytical study of self-avoiding walks on small-world networks, highlighting the continuous increase of the connective constant with rewiring probability.

Findings

Connective constant increases with rewiring probability p.

Linear relation between μ and p for small p.

Different behaviors near p=1 due to connectivity distribution differences.

Abstract

Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was obtained from numerical simulations as a function of the number of steps $n$ on the considered networks. The so-called connective constant, $\mu = \lim_{n \to \infty} u_n/u_{n-1}$, which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability, $p$). For small $p$, one has a linear relation $\mu = \mu_0 + a p$, $\mu_0$ and $a$ being constants dependent on the underlying lattice. Close to $p = 1$ one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small $p$, and differ for $p$ close to 1, because of the different connectivity distributions resulting in both cases.