Topological properties of a time-integrated activity-driven network

Abstract

Here we consider the topological properties of the integrated networks emerging from the activity driven model [Perra at al. Sci. Rep. 2, 469 (2012)], a temporal network model recently proposed to explain the power-law degree distribution empirically observed in many real social networks. By means of a mapping to a hidden variables network model, we provide analytical expressions for the main topological properties of the integrated network, depending on the integration time and the distribution of activity potential characterizing the model. The expressions obtained, exacts in some cases, the results of controlled asymptotic expansions in others, are confirmed by means of extensive numerical simulations. Our analytical approach, which highlights the differences of the model with respect to the empirical observations made in real social networks, can be easily extended to deal with improved, more realistic modifications of the activity driven network paradigm.

Aging and percolation dynamics in a Non-Poissonian temporal network model

“…It is possible to show [35,37] that the degree distribution P t (k) of the resulting network integrated up to time t is functionally related to the probability distribution F (a) from which the activities a i are drawn. Therefore, if fed with the empirically observed F (a), the time-integrated AD networks show some of the topological properties of real social networks, and in particular its characteristic heavy tailed degree distribution [37]. The AD model has proved to be very flexible, allowing to incorporate many typical features of human dynamics, such as memory effects [38], and it is analytically suitable to study dynamical processes on time-varying networks, such as epidemic spreading [39], random walks [20,40], or percolation [41].…”

“…Social activity, which can be defined as the probability per unit time a that an individual becomes active and starts a social interaction, has been empirically measured in a variety of social temporal networks, and shown to exhibit a heterogeneous, heavy tailed distribution [35]. To take into account this heterogeneity, in the AD model [35,37] each node i, representing an agent, is endowed with a constant activity a i , representing the probability that at each time step he/she will establish a link, of infinitesimally short duration, with another agent, chosen uniformly at random. It is possible to show [35,37] that the degree distribution P t (k) of the resulting network integrated up to time t is functionally related to the probability distribution F (a) from which the activities a i are drawn.…”

“…To take into account this heterogeneity, in the AD model [35,37] each node i, representing an agent, is endowed with a constant activity a i , representing the probability that at each time step he/she will establish a link, of infinitesimally short duration, with another agent, chosen uniformly at random. It is possible to show [35,37] that the degree distribution P t (k) of the resulting network integrated up to time t is functionally related to the probability distribution F (a) from which the activities a i are drawn. Therefore, if fed with the empirically observed F (a), the time-integrated AD networks show some of the topological properties of real social networks, and in particular its characteristic heavy tailed degree distribution [37].…”

Control Strategies of Contagion Processes in Time-Varying Networks

“…Moreover, it is possible to prove that integrating activity driven networks in finite time windows such that T N and k N yield graphs characterized by degree distributions following the functional form F(a) [32,82].…”

“…Indeed, spreading processes have been typically considered to take place in either static (τ P τ G ) or annealed (τ P τ G ) networks. While this approximation can be used to study a range of processes such as the spreading of some diseases in contact networks or the propagation of energy in power grids it fails to describe many others phenomena in which the two timescales are comparable [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. In these cases, such as the spreading of ideas, memes, information and some type of diseases the diffusion processes take place in timevarying networks [41,42,43].…”