Network modeling plays a critical role in identifying statistical regularities and structural principles common to many systems. The large majority of recent modeling approaches are connectivity driven. The structural patterns of the network are at the basis of the mechanisms ruling the network formation. Connectivity driven models necessarily provide a time-aggregated representation that may fail to describe the instantaneous and fluctuating dynamics of many networks. We address this challenge by defining the activity potential, a time invariant function characterizing the agents' interactions and constructing an activity driven model capable of encoding the instantaneous time description of the network dynamics. The model provides an explanation of structural features such as the presence of hubs, which simply originate from the heterogeneous activity of agents. Within this framework, highly dynamical networks can be described analytically, allowing a quantitative discussion of the biases induced by the time-aggregated representations in the analysis of dynamical processes. Network modeling [1] [2] [3] [4] [5] [6] has long drawn on the tradition of social network analysis and graph theory, with models ranging from the Erdös-Rényi model to Logit models, p*-models, and Markov random graphs [7] [8] [9] [10] [11] . In the last decade, the class of growing network models, exemplified by the preferential attachment model, has been made widely popular by research in statistical physics and computer science [12] [13] [14] [15] [16] [17] . All these models can be defined as connectivity driven, as the network's topology is at the core of the model's algorithmic definition. Connectivity-driven network models are well-suited for capturing the essential features of systems such as the Internet, where connections among nodes are long-lived elements [18, 19] . However, in many cases the interactions among the elements of the system are rapidly changing and are characterized by processes whose timing and duration are defined on a very short time scale [20, 21] . This limit has been investigated in the case of adaptive systems whose structure evolve being coupled to the process taking place on top of them [22] [23] [24] [25] [26] . Instead, the understating of this limit in time varying networks in which the structure evolves independently of the process is still limited and unexplored. In these activity-driven networks, models intended to capture the process of accumulating connections over time and the resulting degree distribution (i.e. the probability that a node has k connections to other nodes) and other topological properties merely represent a time-integrated perspective of the system. Furthermore, the analysis of dynamical processes in evolving networks is generally performed in the presence of a time-scale separation between the network evolution and the dynamical process unfolding on its structure. In one limit we can consider the network as quenched in its connectivity pattern, thus evolving on a time scale that is much longer that the dynamical process itself. In the other limiting case, the network is evolving at a time scale much shorter than the dynamical process thus effectively disappearing from the definition of the interaction among individuals that is conveniently replaced by effective random couplings. While the time scale separation is extremely convenient for the numerical and analytical tractability of the models, networks generally evolve on a time-scale that might be comparable to the one of the dynamical process [27] [28] [29] [30] [31] [32] . An accurate modelization of the dynamics of activity-driven networks calls for the definition of interaction processes based on the actual measurement of the activity of the agents forming the system, a task now made possible by the availability of time-resolved, high-quality data on the instantaneous activity of millions of agents in a wide variety of networks [33] [34] [35] [36] [37] . Here we present the analysis of three large-scale, time-resolved network datasets and define for each node a measurable quantity, the activity potential, characterizing its interaction pattern within the network. This measure is defined as the number of interactions performed, in a given time window, by each node divided by the total number of interactions made by all the nodes in the same time window. We find that the system level dynamics of the network can be encoded by the activity potential distribution function from which it is possible to derive the appropriate interaction rate among nodes. On the basis of the empirically measured activity potential arXiv:1203.5351v4 [physics.soc-ph]
Perra et al. in REF have provided an activity driven modeling of temporal networks.
Activity driven modeling of time varying networks
Nature Scientific Reports 2, 469 (2012)
Mathematics Computer Science Physics Medicine