Long-tailed distributions of inter-event times as mixtures of exponential distributions

Abstract

Inter-event times of various human behavior are apparently non-Poissonian and obey long-tailed distributions as opposed to exponential distributions, which correspond to Poisson processes. It has been suggested that human individuals may switch between different states in each of which they are regarded to generate events obeying a Poisson process. If this is the case, distributions of inter-event times should approximately obey a mixture of exponential distributions with different parameter values. In the present study, we introduce the minimum description length principle to compare mixtures of exponential distributions with different numbers of components (i.e., constituent exponential distributions). Because these distributions violate the identifiability property, one is mathematically not allowed to apply the Akaike or Bayes information criteria to their maximum likelihood estimator to carry out model selection. We overcome this theoretical barrier by applying a minimum description principle to joint likelihoods of the data and latent variables. We show that mixtures of exponential distributions with a few components are selected as opposed to more complex mixtures in various data sets and that the fitting accuracy is comparable to that of state-of-the-art algorithms to fit powerlaw distributions to data. Our results lend support to Poissonian explanations of apparently non-Poissonian human behavior.1

Metapopulation models imply non-Poissonian statistics of interevent times

“…The results hold true under mild conditions, i.e., for various structures of the metapopulation network, a work-home mobility rule, and higher-order random walks. Although a mixture of exponential distributions is technically not heavy-tailed, it often approximates heavy-tailed distributions reasonably well over scales [35,37,81]. Therefore, the present results provide a compelling explanation of heavy-tailed IET distributions in human and animal contact data.…”

“…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].…”

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

“…A possible extension of this assumption is to the case of more than two states for each node. Then, depending on how such a model translates the nodes' states into the event rate, the distribution of IETs on edges may be approximately a mixture of more than two exponential distributions, which may resemble or actually produce heavy-tailed distributions [36,57,62,63]. In fact, a mixture of a small number of exponential distributions, including the case of just two exponential distributions, is often sufficient for approximating many empirical heavytailed distributions of IETs [36,37,57,62].…”

“…Then, depending on how such a model translates the nodes' states into the event rate, the distribution of IETs on edges may be approximately a mixture of more than two exponential distributions, which may resemble or actually produce heavy-tailed distributions [36,57,62,63]. In fact, a mixture of a small number of exponential distributions, including the case of just two exponential distributions, is often sufficient for approximating many empirical heavytailed distributions of IETs [36,37,57,62]. Therefore, one should carefully assess trade-offs between the complexity of extended models and the explanatory power of the model that one gains by assuming more states for nodes.…”

“…In fact, effects of heavytailed distributions of IETs on epidemic processes [8][9][10][11][12][13][14][15], opinion dynamics [16][17][18][19], evolutionary game dynamics [20], and cascade processes [21][22][23][24], random walks [25][26][27][28], to name a few, have been studied. There are also a number of generative mechanisms and descriptive models * naokimas@buffalo.edu for heavy-tailed distributions of IETs including priority queuing models [6,7,[29][30][31][32][33][34], mixture of exponentials [35][36][37], those supplied by circadian and weekly rhythms [38], and other self-exciting processes [39,40].…”