Fractional dynamics on networks: Emergence of anomalous diffusion and L\'evy flights

TL;DR

This paper introduces a fractional diffusion framework on networks using a fractional Laplacian, enabling long-range random walks that enhance navigation efficiency and produce Le9vy flights, applicable to various network types.

Contribution

The paper develops a spectral-based fractional Laplacian formalism for networks, allowing analysis of anomalous diffusion and navigation with exact results for key properties.

Findings

Fractional random walks improve network navigation efficiency.

Long-range power-law decay leads to Le9vy flights on networks.

The formalism applies to regular, random, and complex networks.

Abstract

We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks with long-range dynamics providing a general framework for anomalous diffusion and navigation, and inducing dynamically the small-world property on any network. We obtained exact results for the stationary probability distribution, the average fractional return probability and a global time, showing that the efficiency to navigate the network is greater if we use a fractional random walk in comparison to a normal random walk. For the case of a ring, we obtain exact analytical results showing that the fractional transition and return probabilities follow a long-range power-law decay, leading to the emergence of L\'evy flights on networks. Our general fractional diffusion formalism applies to regular, random and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.