[Paper Review] Hitting times for non-backtracking random walks.

Dario Fasino, Arianna Tonetto+1 more|arXiv (Cornell University)|May 30, 2021

Complex Network Analysis Techniques25 references4 citations

TL;DR

This paper presents a method to compute the expected hitting times for non-backtracking random walks on finite, undirected graphs by solving a linear system, showing that mean return times for non-backtracking walks exactly match those of standard random walks under the same conditions. The approach enables efficient computation of average first-passage times and round-trip durations.

ABSTRACT

A non-backtracking random walk on a graph is a random walk where, at each step, it is not allowed to return back to the node that has just been left. Non-backtracking random walks can model physical diffusion phenomena in a more realistic way than traditional random walks. However, the interest in these stochastic processes has grown only in recent years and, for this reason, there are still various open questions. In this work, we show how to compute the average number of steps a non-backtracking walker takes to reach a specific node starting from a given node. This problem can be reduced to solving a linear system that, under suitable conditions, has a unique solution. Finally, we compute the average number of steps required to perform a round trip from a given node and we show that mean return times for non-backtracking random walks coincide with their analogue for random walks in all finite, undirected graphs in which both stochastic processes are well-defined.

Motivation & Objective

• To develop a computationally feasible method for calculating the expected number of steps a non-backtracking random walker takes to reach a target node from a given starting node.
• To address the lack of established analytical tools for non-backtracking random walks, which are increasingly relevant for modeling realistic diffusion processes.
• To determine whether mean return times for non-backtracking walks coincide with those of standard random walks in finite, undirected graphs.
• To reduce the problem of computing hitting times to solving a well-defined linear system under suitable conditions.

Proposed method

• Model the non-backtracking random walk as a Markov process on the directed edges of the graph, avoiding immediate reversals.
• Formulate the expected hitting time to a target node as the solution to a linear system derived from the transition probabilities of the non-backtracking process.
• Ensure the linear system has a unique solution under standard graph-theoretic conditions, such as connectivity and finite size.
• Use the structure of the graph and the non-backtracking transition rules to define the coefficient matrix of the system.
• Derive the mean return time by computing the expected time to return to the starting node after leaving it, using the same linear system framework.
• Leverage symmetry and graph structure to compare results with standard random walks and verify equivalence in mean return times.

Experimental results

Research questions

• RQ1 How can the expected hitting time of a non-backtracking random walk be computed efficiently for a given graph and target node?
• RQ2 Under what conditions does the linear system for non-backtracking hitting times have a unique solution?
• RQ3 Do the mean return times for non-backtracking random walks match those of standard random walks in finite, undirected graphs?
• RQ4 Can the hitting time computation be reduced to a solvable linear system without requiring simulation or approximation?

Key findings

• The expected hitting time for a non-backtracking random walk to reach a specific node from a starting node can be computed by solving a linear system that is guaranteed to have a unique solution under standard graph conditions.
• The mean return time for non-backtracking random walks is identical to that of standard random walks in all finite, undirected graphs where both processes are well-defined.
• The linear system formulation allows for exact computation of average hitting times without relying on Monte Carlo simulations or asymptotic approximations.
• The equivalence of mean return times between non-backtracking and standard random walks holds universally across all finite, undirected graphs with well-defined transition rules.
• The method provides a computationally efficient and analytically rigorous alternative to simulation-based estimation of first-passage times in non-backtracking processes.