Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks

TL;DR

This paper derives analytical formulas for the distribution of first return times of non-backtracking random walks on various network models, revealing how network structure influences dynamic processes.

Contribution

It provides the first analytical expressions for first return time distributions of NBWs on configuration model networks, linking structural properties to dynamical behavior.

Findings

Tail distribution of first return times expressed via degree distribution

Mean first return time matches Kac's lemma for simple RWs

Closed-form formulas for ER, regular, exponential, and power-law networks

Abstract

We present analytical results for the distribution of first return (FR) times of non-backtracking random walks (NBWs) on undirected configuration model networks consisting of $N$ nodes with degree distribution $P(k)$. We focus on the case in which the network consists of a single connected component. Starting from a random initial node $i$ at time $t=0$, an NBW hops into a random neighbor of $i$ at time $t=1$ and at each subsequent step it continues to hop into a random neighbor of its current node, excluding the previous node. We calculate the tail distribution $P ( T_{\rm FR} > t )$ of first return times from a random initial node to itself. It is found that $P ( T_{\rm FR} > t )$ is given by a discrete Laplace transform of the degree distribution $P(k)$. This result exemplifies the relation between structural properties of a network, captured by the degree distribution, and properties of dynamical processes taking place on the network. Using the tail-sum formula, we calculate the mean first return time ${\mathbb E}[ T_{\rm FR} ]$. Surprisingly, ${\mathbb E}[ T_{\rm FR} ]$ coincides with the result obtained from Kac's lemma that applies to simple random walks (RWs). We also calculate the variance ${\rm Var}(T_{\rm FR})$, which accounts for the variability of first return times between different NBW trajectories. We apply this formalism to Erd{\H o}s-R\'enyi networks, random regular graphs and configuration model networks with exponential and power-law degree distributions and obtain closed-form expressions for $P( T_{\rm FR} > t )$ as well as its mean and variance. These results provide useful insight on the advantages of NBWs over simple RWs in network exploration, sampling and search processes.