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

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-Rényi 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.

💡 Research Summary

This paper presents a comprehensive analytical treatment of the first‑return time (FRT) statistics for non‑backtracking random walks (NBWs) on undirected configuration‑model networks. The authors restrict attention to ensembles in which the entire graph forms a single giant component, thereby avoiding isolated nodes and leaf vertices. An NBW starts at a randomly chosen node i at time t = 0, moves to a random neighbor at t = 1, and thereafter at each step selects uniformly among all neighbors except the node visited in the previous step. Because backtracking is prohibited, an NBW on an infinite tree can never return to its origin; in a finite graph a return is only possible via a trajectory that contains at least one cycle, so the minimal return time is three steps.

The core of the analysis is the derivation of the tail distribution (P(T_{\rm FR}>t)), where (T_{\rm FR}) denotes the first time the walk revisits its starting node. Conditioning on the degree (K=k) of the initial node, the authors construct a recursive relation for the conditional tail probability. They show that the probability of not having returned after t steps can be expressed as (