Burstiness and fractional diffusion on complex networks

TL;DR

This paper investigates how fat-tailed inter-event time distributions with diverging averages affect random walk dynamics on networks, revealing universal power-law relaxation behaviors described by fractional derivatives.

Contribution

It introduces a fractional calculus framework to analyze the impact of heavy-tailed waiting times on network dynamics, highlighting the absence of time-scale separation among modes.

Findings

All dynamical modes exhibit the same power-law relaxation asymptotically.

The dynamics lack time-scale separation, making all modes equally relevant over time.

Numerical simulations confirm the theoretical predictions.

Abstract

Many dynamical processes on real world networks display complex temporal patterns as, for instance, a fat-tailed distribution of inter-events times, leading to heterogeneous waiting times between events. In this work, we focus on distributions whose average inter-event time diverges, and study its impact on the dynamics of random walkers on networks. The process can naturally be described, in the long time limit, in terms of Riemann-Liouville fractional derivatives. We show that all the dynamical modes possess, in the asymptotic regime, the same power law relaxation, which implies that the dynamics does not exhibit time-scale separation between modes, and that no mode can be neglected versus another one, even for long times. Our results are then confirmed by numerical simulations.