Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights

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 Lévy 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.

8 pages, 4 figures

References in corpus (9)

• Understanding individual human mobility patterns
• Synchronization in complex networks
• The scaling laws of human travel
• Diffusion dynamics on multiplex networks
• Fractional Laplacian in Bounded Domains
• Random walks and search in time-varying networks
• Long-Range Navigation on Complex Networks using Lévy Random Walks
• Efficiency of quantum and classical transport on graphs
• Evolution of optimal Lévy-flight strategies in human mental searches

Cited by in corpus (9)

• Fractional random walk lattice dynamics
• On recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
• Biased continuous-time random walks with Mittag-Leffler jumps
• A fractional generalization of the classical lattice dynamics approach
• Fractional combinatorial Calabi flow on surfaces
• Navigation by anomalous random walks on complex networks
• Network navigation using Page Rank random walks
• Markov chain approach to anomalous diffusion on Newman-Watts networks
• RaWaNet: Enriching Graph Neural Network Input via Random Walks on Graphs