Onset of anomalous diffusion from local motion rules

TL;DR

This paper introduces a local rule-based model for anomalous diffusion on a 1D lattice, showing how cascades of steps and power-law waiting times can produce superdiffusive behavior, with potential applications in complex and temporal networks.

Contribution

The work presents a novel local mechanism for anomalous diffusion using cascades of steps and power-law waiting times, expanding understanding beyond traditional Lévy flights.

Findings

Power-law distributed cascades induce superdiffusion or normal diffusion.

Waiting times with power-law distribution influence scaling behavior.

Model applicable to complex and temporal networks.

Abstract

Anomalous diffusion processes, in particular superdiffusive ones, are known to be efficient strategies for searching and navigation by animals and also in human mobility. One way to create such regimes are L\'evy flights, where the walkers are allowed to perform jumps, the "flights", that can eventually be very long as their length distribution is asymptotically power-law distributed. In our work, we present a model in which walkers are allowed to perform, on a 1D lattice, "cascades" of $n$ unitary steps instead of one jump of a randomly generated length, as in the L\'evy case, where $n$ is drawn from a cascade distribution $p_n$. We show that this local mechanism may give rise to superdiffusion or normal diffusion when $p_n$ is distributed as a power law. We also introduce waiting times that are power-law distributed as well and therefore the probability distribution scaling is steered by the two PDF's power-law exponents. As a perspective, our approach may engender a possible generalization of anomalous diffusion in context where distances are difficult to define, as in the case of complex networks, and also provide an interesting model for diffusion in temporal networks.