Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks

Martin Gueuning,Jean-Charles Delvenne,Renaud Lambiotte

doi:10.3390/e19100542

Martin Gueuning, Jean-Charles Delvenne + Show 1 more

Open Access

https://doi.org/10.3390/e19100542

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Abstract

We consider the problem of diffusion on temporal networks, where the dynamics of each edge is modelled by an independent renewal process. Despite the apparent simplicity of the model, the trajectories of a random walker exhibit non-trivial properties. Here, we quantify the walker’s tendency to backtrack at each step (return where he/she comes from), as well as the resulting effect on the mixing rate of the process. As we show through empirical data, non-Poisson dynamics may significantly slow down diffusion due to backtracking, by a mechanism intrinsically different from the standard bus paradox and related temporal mechanisms. We conclude by discussing the implications of our work for the interpretation of results generated by null models of temporal networks.

Highlights

Random walks (RWs) play a key role in network theory [1]
The paradox lies in the fact that the random walker has a tendency to take one particular edge over the others, even if each edge is statistically equivalent
We have shown that the shape of the inter-activation distribution may induce a backtracking bias for random walkers on a temporal network

Summary

Introduction

Random walks (RWs) play a key role in network theory [1]. RWs are at the core of algorithms to explore the network structure and to uncover its important features, such as the centrality of the nodes [2,3]) or the presence of communities and modules [4,5]. A central question is to understand the mechanisms that either accelerate or slow down the diffusion, for instance through the characteristic time for the dynamics to converge to the equilibrium state This question has been considered by means of numerical simulations, by simulating a diffusive process on empirical temporal network data [14], and comparing its speed with the same process run on randomized null models [15]. We estimate the impact of the resulting bias to backtrack on the mixing rate of the process Taken together, these results allow to quantify a mechanism that may either slow down or accelerate diffusion, by changing the number of steps leading to mixing, which is inherently different from well-known mechanisms such as the bus paradox [19] or other temporal mechanisms [20], only affecting the time to relaxation, and not the number of steps. Our observations allow us to gain insight into unexpected properties of a standard null model for temporal network analysis

Bias on the Probability of Backtracking

Impact of Backtracking on the Mixing Rate of the Random Walk

Discussion