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

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. 1. Introduction. Few mathematical models have found so many applications in the physical, chemical, biological, social and economical sciences as the random walk model [20, 27]. The term " random walk " was first proposed by K. Pearson [38] in an informal question posted in Nature in 1905. Pearsons description reads as " A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. " Among the first respondents to Pearson, Lord Rayleigh [40] already pointed out a connection existing between random walks and other physical processes, namely that a random walk " is the same as that of the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at random ". Connections like these between random walks and many physico-chemical, biological and socioeconomic processes are what guarantees the great vitality of this research topic. This includes relations between random walks, Brownian motion and diffusive processes in general [9], the connection between random walks and the classical theory of electricity [10, 34], and the formulation of the efficient market hypothesis [26], among others. In order to distinguish the originally-proposed random walk model …