Burstiness and Aging in Social Temporal Networks

Abstract

The presence of burstiness in temporal social networks, revealed by a power-law form of the waiting time distribution of consecutive interactions, is expected to produce aging effects in the corresponding time-integrated network. Here, we propose an analytically tractable model, in which interactions among the agents are ruled by a renewal process, that is able to reproduce this aging behavior. We develop an analytic solution for the topological properties of the integrated network produced by the model, finding that the time translation invariance of the degree distribution is broken. We validate our predictions against numerical simulations, and we check for the presence of aging effects in a empirical temporal network, ruled by bursty social interactions. DOI: 10.1103/PhysRevLett.114.108701 PACS numbers: 89.75.Hc Our understanding of the structure and properties of social interactions has experienced a boost in recent years due to the new availability of large amounts of digital empirical data [1,2]. This endeavor has found the necessary theoretical grounding in network science [3,4]. A first round of studies focused on a static network representation [5,6], in which nodes (standing for individuals) and edges (indicating social interactions) are constant and never change in time. From such a static representation, a wealth of complex topological information was obtained, concerning, e.g., the presence of scale-free, power-law degree distributions PðkÞ ∼ k −γ , large clustering, positive degree correlations, or a distinct community structure [7]. More recently, this framework has been challenged by the empirical observation of a temporal dimension in networks, particularly evident in social systems, due to the fact that social relationships are continuously created and terminated. From these temporal networks [8], a static representation is obtained by means of an integration of the instantaneous interactions over a time window of width t, and its associated topological properties, such as the degree distribution P t ðkÞ, are thus to be understood to depend on the integration time [9]. The empirical study of the temporal aspects of social networks has unveiled an additional level of complexity, embodied in many statistical properties showing heavy-tailed distributions. Remarkable among them are the distribution ψðτÞ of interevent or waiting times between two consecutive social interactions, revealing the bursty nature of human dynamics, and the distribution FðaÞ of social activity, measuring the probability per unit time of establishing a new social relation. Both distributions approximately obey power-law decays of the form ψðτÞ ∼ τ −ð1þαÞ and FðaÞ ∼ a −δ , respectively [10][11][12][13][14]. Noteworthy, the bursty nature of social interactions represents a common feature of human dynamics that can be related to the prioritized behavior of human beings [11]. This twofold nature of social interactions naturally raises the issue of the relation between the temporal correlation properties of time-varying netw...

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

“…(13), we arrive at [27] it was argued that it is directly related to the average activityā, defined as the probability to become active in a time window of a given length ∆t. Given the power law activity distribution observed in real temporal networks [35], here we assume a distribution…”

“…For the standard AD model, with a i constant, we have ψ AD i (τ ) = a i e −aiτ , that is, agents follow a simple Poisson process [42]. Any function a i (τ ) leads thus to the consideration of a generalized non-Poissonan activity driven (NoPAD) model [27]. Shifting away from the instantaneous activity a i (τ ), the NoPAD model can be defined in terms of a set of agents that become active by following a renewal process with waiting time distribution ψ i (τ ), giving the probability of observing a time τ between two activation events of agent i.…”

“…Among those, it is noteworthy the recently proposed Non-Poissoinan activity driven (NoPAD) model [27]. The NoPAD model aims to be a generative model, akin to the configuration model for static networks [28].…”

Generalization of Pairwise Models to non-Markovian Epidemics on Networks

“…However, there is renewed interest in non-Markovian processes, such as epidemics on networks [12][13][14][15][16][17][18], random walks [19], and temporal networks [20]. For example, Min et al [13] consider the SIR model with fixed recovery and an infectious process with heavy-tail distribution.…”

“…Examples include intertrade durations in financial markets [21], socionetworks [22], or contacts between individuals being dynamic [20]. In the context of epidemiology, the period of infectiousness has a key role [23,24].…”