Random walk on temporal networks with lasting edges

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.

Rock–paper–scissors dynamics from random walks on temporal multiplex networks

“…We study the impact of the precedence on an active random walk, however it is worth noticing that its impact on the passive random walk may lead to opposite bias towards layers. Indeed, in the presence of short cycles for the passive walker, an additional competition will emerge induced by the short cycles [27]. For non-exponential distributions, this effect results in a backtracking bias towards or against the last traveled edges [28], typically leading to the emergence of short cycle patterns in human-related network [29].…”

Classes of random walks on temporal networks with competing timescales

“…Moreover, edge j → i needs to activate exactly after the duration t − x. With this in mind, and when all distributions are exponential, it was shown in [18] that :…”

“…We have r = λ λ+η and 1 − r = η λ+η . It was therefore shown in [18] that T i j (t ) has two terms, such that the transition density from node j reads…”

“…Let also q * (s) = 1 − p * (s) be the probability the edge is available for transport. These two quantities were computed in [18], by accounting for all possible on-off switches of the edge in the interval [0, s]. The resulting expression has a strikingly simple form when U (t ) and D(t ) have the same (exponential density) rate η = λ, our working hypothesis in what follows :…”

“…• A jump across an edge some time in the past (meaning it was available back then) increases the probability that the same edge is again available to the walker who has returned on the node after a cycle, and is ready for another jump. This memory that impacts the state of the edge was described in details in [18], and is captured by the duration-dependent probability p * in equation (19).…”