The random walk process underlies the description of a large number of real world phenomena. Here we provide the study of random walk processes in time varying networks in the regime of timescale mixing; i.e. when the network connectivity pattern and the random walk process dynamics are unfolding on the same time scale. We consider a model for time varying networks created from the activity potential of the nodes, and derive solutions of the asymptotic behavior of random walks and the mean first passage time in undirected and directed networks. Our findings show striking differences with respect to the well known results obtained in quenched and annealed networks, emphasizing the effects of dynamical connectivity patterns in the definition of proper strategies for search, retrieval and diffusion processes in time-varying networks. PACS numbers: 89.75.Hc, 05.40.Fb Random walks on networks lie at the core of many real-world dynamical processes, ranging from the navigation and ranking of information networks to the spreading of diseases and the routing of information packets in large-scale infrastructures such as the Internet [1] [2] [3] [4] [5] [6] . In recent years, empirical evidence pointing out the heterogeneous topology of many real-world networks has led to a large body of work focusing on the properties of random walks in networks characterized by heavy tailed degree distributions and other features such as clustering and community structure [1, [7] [8] [9] [10] [11] [12] . Although these studies provided a deeper understanding of processes of technological relevance such as WWW navigation and ranking, they have mostly focused on the situation in which the time scale characterizing the changes in the structure of the network and the time scale describing the evolution of the process are well separated [13] [14] [15] [16] [17] . While convenient for analytical tractability, this limit is far from realistic in many systems including modern information networks, the diffusion and search of information in microblogging systems and social networking platforms, sexually transmitted diseases or the diffusion of ideas and knowledge in social contexts. In all these cases, the concurrence of contacts and their dynamical patterns are typically characterized by a time scale comparable to that of the diffusion process, motivating the study models able to account for effects of the time varying nature of networks on dynamical processes [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] . Motivated by the above problems we study the random walk process in a fairly general class of timevarying networks. Namely, we consider the activity driven class of models for time-varying networks presented in Ref. [23] that allows for an explicit representation of dynamical connectivity patterns. We derive the analytical solutions of the stationary state of the random walk and the mean first passage time [28] in both directed and undirected time varying networks. We find that the behavior of the random walk and the ensuing network discovery process in time varying networks is strikingly different from those occurring in quenched and annealed topologies [1, 3, 8, 29] . These results have the potential to become a starting point for the definition of alternative strategies and mechanisms to explore and retrieve information from networks, and characterize more accurately spreading and diffusion processes in a wide range of dynamic social networks. Activity driven network models focus on the activity pattern of each node, which is used to explicitly model the evolution of the connectivity pattern over time. Each node i is characterized by a quenched, fixed, activity rate a i , extracted from a distribution F (a), that represents the probability per unit time that a given node will engage in an interaction and generate the corresponding edge connecting it with other nodes in the system. In the simplest formulation of the model, networks are generated according to the following memoryless stochastic process [23]: i) At each time step t, the instantaneous network G t starts with N disconnected nodes; ii) With probability a i ∆t, each vertex i becomes active and generates m undirected links that are connected to m other randomly selected vertices. Non-active nodes can still receive connections from other active vertices; iii) At time t + ∆t, all the edges in the network G t are deleted and the process starts over again to generate the network G t+∆t . It can be shown that the full dynamics of the network is encoded in the activity rate distribution, F (a), and that the time-aggregated measurement of network connectivity yields a degree distribution that follows the same functional form as the distribution F (a). This distribution can be assumed a priori or derived from empirical data in the case of high quality data set
Perra et al. studied the effect of timevarying networks in random walks and search processes REF .
Random walks and search in time-varying networks
Physical Review Letters, 109, 238701 (2012)
Medicine Physics