Generative models of simultaneously heavy-tailed distributions of interevent times on nodes and edges

Abstract

Intervals between discrete events representing human activities, as well as other types of events, often obey heavy-tailed distributions, and their impacts on collective dynamics on networks such as contagion processes have been intensively studied. The literature supports that such heavy-tailed distributions are present for inter-event times associated with both individual nodes and individual edges in networks. However, the simultaneous presence of heavy-tailed distributions of inter-event times for nodes and edges is a non-trivial phenomenon, and its origin has been elusive. In the present study, we propose a generative model and its variants to explain this phenomenon. We assume that each node independently transits between a high-activity and low-activity state according to a continuous-time two-state Markov process and that, for the main model, events on an edge occur at a high rate if and only if both end nodes of the edge are in the high-activity state. In other words, two nodes interact frequently only when both nodes prefer to interact with others. The model produces distributions of inter-event times for both individual nodes and edges that resemble heavy-tailed distributions across some scales. It also produces positive correlation in consecutive inter-event times, which is another stylized observation for empirical data of human activity. We expect that our modeling framework provides a useful benchmark for investigating dynamics on temporal networks driven by non-Poissonian event sequences.

Metapopulation models imply non-Poissonian statistics of interevent times

“…Therefore, the present results provide a compelling explanation of heavy-tailed IET distributions in human and animal contact data. Additional mechanisms such as circadian or weekly rhythms [33] and dynamics of individuals' internal states [39] on top of mobility and metapopulation networks may make IET distributions more smooth and more power-law-like.…”

“…In most (but not all) cases, heavy-tailed IET distributions slow down contagion and diffusion in epidemic processes [7][8][9][10][11][12][13][14], opinion dynamics [15][16][17][18], evolutionary game dynamics [19], cascade processes [20][21][22][23], and random walks [24][25][26][27]. Several mechanisms can generate heavy-tailed IET distributions, including priority queue models [5,6,[28][29][30][31][32], self-exciting processes [33][34][35], mixture of exponentials [36][37][38], and dynamics of nodal states, where mutual agreement of two nodes produces contact events at a high rate [39].…”

“…However, these models do not explain the genesis of heavy-tailed IET distributions present in one dominant case of human (and animal) activity-mobility. In fact, most of face-to-face contact or proximity data have been collected from mobile individuals, and such data show heavy-tailed IET distributions [25,[39][40][41][42][43][44][45][46][47]. Mobility constrains the statistics of IETs because contact events may occur only when individuals are copresent in the same place.…”

Effects of concurrency on epidemic spreading in Markovian temporal networks

“…2. The model resembles the so-called AND model in [33]. An intuitive interpretation of model 2 is that two individuals chat with each other if and only if both of them want to.…”

“…2 for a schematic). This model resembles the OR model proposed in [33]. The intuition behind model 3 is that one person can start conversation with another person whenever either person wants to talk regardless of whether the other person wants to.…”