Slow relaxation dynamics and aging in random walks on activity driven temporal networks

The heterogeneous topology of a complex network can have a very relevant impact on the properties of dynamical systems running on top of it. Already classical studies in network science have thus shown that a heterogeneous connectivity pattern can lead to a null percolation threshold, set a strong resilience against random failures, as well as to induce a vanishing epidemic threshold for disease propagation. Similar and additional remarkable effects have been observed in a wide variety of dynamical processes.
Such dynamical effects, originally reported for static networks, in which nodes and edges are fixed and do not change over time, can take a different, more complex turn when one considers the intrinsic time-varying, temporal nature of many real networks. Indeed, networked systems are often not static, but show connections which appear and disappear with some characteristic time scales. Social networks represent the prototypical example of this behavior, being defined in terms of a sequence of social contacts that are continuously established and broken. This mixing of time scales can induce new phenomenology on dynamics on temporal networks, in stark contrast with what is observed in static networks. Moreover, the bursty nature of the time evolution of temporal network contacts, characterized by long stretches of inactivity, interspersed by bursts of intense activity, can complicate the picture, inducing for example a dynamical slowing down in dynamical processes as varied as epidemic spreading, diffusion or synchronization. The random walk is one of the simplest dynamical processes, although still underlying many practical realistic applications such as diffusion, searching, community detection and spreading dynamics. Even in this simplest of cases, a time-varying substrate can induce very noticeable differences with respect to the behavior expected in static networks. In this work, we investigate the dynamic relaxation of random walks on temporal networks by focusing in the recently proposed activity driven model [N. Perra, {\it et. al.} Sci.Rep. 2, 469 (2012)]. For realistic activity distributions with a power-law form, we observe the presence of a very slow relaxation dynamics compatible with aging effects. A theoretical description of this processes in achieved by means of a mapping to Bouchaud?s trap model. The mapping highlights the profound difference in the dynamics of the random walks according to the value of the exponent $\gamma$ in the activity distribution.
Acknowledgements and Financial Support: FAPEMIG, CAPES and CNPq.