Random walks with long-range steps generated by functions of Laplacian matrices

TL;DR

This paper investigates non-local random walk strategies on networks using functions of Laplacian matrices, revealing two main types of long-range behaviors: Brownian motion and Lévy flights, with implications for network exploration.

Contribution

It introduces a generalized framework for random walks based on Laplacian functions, characterizing conditions for Brownian and Lévy flight behaviors in complex networks.

Findings

Only two Laplacian function types produce distinct long-range behaviors.

Long-range step behavior depends on the lowest non-vanishing order of the Laplacian function.

The framework applies to various network types, including lattices, trees, and complex networks.

Abstract

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions L\'evy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions. In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like L\'evy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks.