The distribution of path lengths of self avoiding walks on Erd\H{o}s-R\'enyi networks

TL;DR

This paper analyzes the length distribution of self avoiding walks on Erdős-Rényi networks, revealing that the path lengths follow a Gompertz distribution and providing both analytical and numerical insights into their behavior.

Contribution

The study introduces an analytical framework for the path-length distribution of SAWs on Erdős-Rényi networks, showing it follows a Gompertz distribution and detailing the pruning effect on network degree distribution.

Findings

Path-length distribution follows Gompertz distribution.

Pruned networks maintain Poisson degree distribution with decreasing mean.

Number of SAW paths increases dramatically with path length.

Abstract

We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erd\H{o}s-R\'enyi networks we show that the pruned networks maintain a Poisson degree distribution, $p_t(k)$, with an average degree, $\langle k \rangle_t$, that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, $n_T(\ell)$, increases dramatically as a function of $\ell$. We also obtain analytical results for the path-length distribution, $P(\ell)$, of the SAW paths which are actually pursued, starting from a random initial node. It turns out that $P(\ell)$ follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.