The distribution of first hitting times of non-backtracking random walks on Erd\H{o}s-R\'enyi networks

TL;DR

This paper analytically characterizes the distribution of first hitting times for non-backtracking random walks on Erdős-Rényi networks, revealing how network density influences walk termination mechanisms and path lengths.

Contribution

It provides the first analytical derivation of the tail distribution of first hitting times for non-backtracking walks on Erdős-Rényi networks, including effects of trapping and retracing.

Findings

Distribution of first hitting times is a product of Rayleigh and exponential distributions.

Paths are longer than simple random walks but shorter than self-avoiding walks.

Trapping dominates in sparse networks, retracing in dense networks.

Abstract

We present analytical results for the distribution of first hitting times of non-backtracking random walks on finite Erd\H{o}s-R\'enyi networks of $N$ nodes. The walkers hop randomly between adjacent nodes on the network, without stepping back to the previous node, until they hit a node which they have already visited before or get trapped in a dead-end node. At this point, the path is terminated. The length, $d$, of the resulting path, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,\dots$, of non-backtracking random walks starting from a random initial node. It turns out that the distribution $P(d>\ell)$ is given by a product of a discrete Rayleigh distribution and an exponential distribution. It is found that the paths of non-backtracking random walks, up to their termination at the first hitting time, are longer, on average, than those of the corresponding simple RWs. However, they are shorter than those of SAWs on the same network, which terminate at the last hitting time. We obtain analytical results for the probabilities, $p_{\rm ret}$ and $p_{\rm trap}$, that a path will terminate by retracing, namely stepping into an already visited node, or by trapping, namely entering a node of degree $k=1$, which has no exit link, respectively. It is shown that in dilute networks the dominant termination scenario is trapping while in dense networks most paths terminate by retracing. We obtain expressions for the conditional tail distributions of path lengths, $P(d>\ell|{\rm ret})$ and $P(d>\ell|\rm{trap})$, for those paths which terminate by retracing or by trapping, respectively. We also study a class of generalized non-backtracking random walk models which not only avoid the backtracking step into the previous node but avoid stepping into the last $S$ visited nodes.