Random walk on temporal networks with lasting edges

TL;DR

This paper develops a comprehensive analytical framework to study random walks on temporal networks with edges that appear and disappear over time, revealing non-Markovian behaviors even with memoryless processes.

Contribution

It introduces a novel analytical approach for non-Markovian random walks on temporal networks with cycles, extending previous models limited to acyclic graphs.

Findings

Non-Markovian trajectories can emerge in cyclic networks with memoryless processes.

The framework accurately characterizes walk dynamics on directed acyclic and cyclic graphs.

Numerical simulations validate the analytical predictions.

Abstract

We consider random walks on dynamical networks where edges appear and disappear during finite time intervals. The process is grounded on three independent stochastic processes determining the walker's waiting-time, the up-time and down-time of edges activation. We first propose a comprehensive analytical and numerical treatment on directed acyclic graphs. Once cycles are allowed in the network, non-Markovian trajectories may emerge, remarkably even if the walker and the evolution of the network edges are governed by memoryless Poisson processes. We then introduce a general analytical framework to characterize such non-Markovian walks and validate our findings with numerical simulations.