# Random walks in non-Poissoinan activity driven temporal networks

## TL;DR

This paper investigates how non-Poissonian, bursty temporal network dynamics influence random walk processes, revealing fundamental differences from Markovian networks and surprising behaviors in the infinite average inter-event time limit.

## Contribution

It provides analytical expressions for steady state and first passage times of random walks on non-Markovian temporal networks, highlighting unique behaviors absent in Markovian cases.

## Key findings

- Random walks on non-Poissonian networks differ from Markovian cases.
- In the limit of diverging inter-event times, the network appears homogeneous to the walker.
- Numerical simulations support the analytical results.

## Abstract

The interest in non-Markovian dynamics within the complex systems community has recently blossomed, due to a new wealth of time-resolved data pointing out the bursty dynamics of many natural and human interactions, manifested in an inter-event time between consecutive interactions showing a heavy-tailed distribution. In particular, empirical data has shown that the bursty dynamics of temporal networks can have deep consequences on the behavior of the dynamical processes running on top of them. Here, we study the case of random walks, as a paradigm of diffusive processes, unfolding on temporal networks generated by a non-Poissonian activity driven dynamics. We derive analytic expressions for the steady state occupation probability and first passage time distribution in the infinite network size and strong aging limits, showing that the random walk dynamics on non-Markovian networks are fundamentally different from what is observed in Markovian networks. We found a particularly surprising behavior in the limit of diverging average inter-event time, in which the random walker feels the network as homogeneous, even though the activation probability of nodes is heterogeneously distributed. Our results are supported by extensive numerical simulations. We anticipate that our findings may be of interest among the researchers studying non-Markovian dynamics of time-evolving complex topologies.