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

TL;DR

This paper analytically characterizes the distribution of first hitting times for random walks on Erd ext{o}s-Rényi networks, revealing how network density influences termination scenarios and path length distributions.

Contribution

It introduces recursive equations to derive the tail distribution of path lengths and termination probabilities, validated by simulations, with insights into different termination mechanisms.

Findings

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

In dilute networks, backtracking dominates as the termination cause.

In dense networks, retracing is the primary termination scenario.

Abstract

Analytical results for the distribution of first hitting times of random walks on Erd\H{o}s-R\'enyi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has already visited before. At this point, the path terminates. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, $P(d > \ell)$. The results are found to be in excellent agreement with numerical simulations. It is found %turns out that the distribution $P(d > \ell)$ follows a product of an exponential distribution and a Rayleigh distribution. The mean, median and standard deviation of this distribution are also calculated, in terms of the network size and its mean degree. The termination of an RW path may take place either by backtracking to the previous node or by retracing of its path, namely stepping into a node which has been visited two or more time steps earlier. We obtain analytical results for the probabilities, $p_b$ and $p_r$, that the cause of termination will be backtracking or retracing, respectively. It is shown that in dilute networks the dominant termination scenario is backtracking while in dense networks most paths terminate by retracing. We also obtain expressions for the conditional distributions $P(d=\ell | b)$ and $P(d=\ell | r)$, for those paths which are terminated by backtracking or by retracing, respectively. These results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.