Random Multi-Hopper Model. Super-Fast Random Walks on Graphs

TL;DR

This paper introduces a multi-hopper random walk model on graphs that uses long-range jumps with decaying probabilities, significantly reducing hitting times and enhancing exploration efficiency compared to standard random walks.

Contribution

The paper presents a novel multi-hopper model with long-range jumps that accelerates graph exploration, approaching the speed of a complete graph as parameters tend to zero.

Findings

Multi-hopper's hitting times converge to minimal values similar to a complete graph.

The model explores clustered and skewed networks more efficiently.

Computational experiments confirm significant speed-up over normal random walks.

Abstract

We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multi-hopper's hitting times between any two nodes in the graph converge to their minimum possible value, given by the hitting times of a normal random walker on a complete graph. Stated differently, for small parameter values the multi-hopper explores a general graph as fast as possible when compared to a random walker on a full graph. Using computational experiments we show that compared to the normal random walker, the multi-hopper indeed explores graphs with clusters or skewed degree distributions more efficiently for a large parameter range. We provide further computational evidence of the speed-up attained by the random multi-hopper model with respect to the normal random walker by studying deterministic, random and real-world networks.