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

Dor Lev-Ari; Ido Tishby; Ofer Biham; Eytan Katzav; Diego Krapf

Paper

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

nodes with degree distribution

. We focus on the case in which the network consists of a single connected component. Starting from a random initial node

at time

, an NBW hops into a random neighbor of

at time

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

of first return times from a random initial node to itself. It is found that

is given by a discrete Laplace transform of the degree distribution

. 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

. Surprisingly,

coincides with the result obtained from Kac's lemma that applies to simple random walks (RWs). We also calculate the variance

, which accounts for the variability of first return times between different NBW trajectories. We apply this formalism to Erd{\H o}s-Rényi networks, random regular graphs and configuration model networks with exponential and power-law degree distributions and obtain closed-form expressions for

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.