Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks

Abstract

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.

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

“…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].…”

“…The walk is called active since waiting times are drawn once the walker arrives on a node, as opposite to the passive walk where contacts are taking place on edges regardless of the presence of a random walker. Note that, when the network is directed and has no short cycles, a passive walk may be approximated by an active one provided that waiting time distributions are adapted accordingly [28,1]. Because only the first edge to reach activation is taken by the walker, there is a underlying competition between different edges.…”

Continuous-Time Random Walks and Temporal Networks

“…A good example concerns the use of null models to determine how the temporal nature of a realworld network affects diffusion. A popular solution consists in comparing numerical simulations of diffusion on the original data and on different versions of randomised data [31]. The results from section IV B show that randomised null models in which temporal correlations are removed yet have the tendency towards backtracking, and thus to slow down the exploration of the network.…”

Random walk on temporal networks with lasting edges

“…This assumption is clearly valid for DAGs but ceases to hold true when the underlying network has cycles. In that case, the walker may be influenced by the statistical information left at the previous passage, which may induced biases in the walker trajectory [23,24]. The acyclic predictions are however expected to remain good approximations if the process on the nodes (ψ) is slow with respect to the edges dynamics, either in the case of long cycles or also locally if nodes have high degree.…”

“…When the dynamics of the edges is Poissonian, trajectories are encoded by a Markov chain, whereas the timings obtained from a non-Poisson renewal process lead to non-trivial properties such as the emergence of non-Markovian trajectories. In that case the trajectory of the walker generally depends on its * julien.petit@unamur.be previous trajectory and not only on its current location [23,24]. The emergence of non-Markovian trajectories is even more pronounced in situations when the activations of edges are correlated, often requiring the use of higherorder models for the data [25,26].…”