Patterns of deliberate human activity and behavior are of utmost importance in areas as diverse as disease spread, resource allocation, and emergency response. Because of its widespread availability and use, e-mail correspondence provides an attractive proxy for studying human activity. Recently, it was reported that the probability density for the inter-event time τ between consecutively sent e-mails decays asymptotically as τ −α , with α ≈ 1. The slower than exponential decay of the inter-event time distribution suggests that deliberate human activity is inherently nonPoissonian. Here, we demonstrate that the approximate powerlaw scaling of the inter-event time distribution is a consequence of circadian and weekly cycles of human activity. We propose a cascading non-homogeneous Poisson process which explicitly integrates these periodic patterns in activity with an individual's tendency to continue participating in an activity. Using standard statistical techniques, we show that our model is consistent with the empirical data. Our findings may also provide insight into the origins of heavy-tailed distributions in other complex systems. human activity | point process | hypothesis testing | complex systems T he analysis of social and economic data has a long and illustrious history [1, 2, 3] . Despite their idiosyncratic complexity, a number of striking statistical regularities are known to describe individual and societal human behavior [4, 5, 6, 7] . These regularities are of enormous practical importance because they provide insight into how individual behaviors influence social and economic outcomes. Indeed, much of the current research on complex systems aims to quantify the impact of individual agents on the organization and dynamics of the system as a whole [8, 9] . Before we can predict how individuals affect, for example, the organization of systems, it is paramount to understand the behavior of the individual agents. The current availability of digital records has made it much easier for researchers to quantitatively investigate various aspects of human behavior [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] . In particular, e-mail communication records are attracting much attention as a proxy for quantifying deliberate human behavior due to the omnipresence of e-mail communication and availability of e-mail records [13, 14, 18, 16] . The data, however, does not provide a detailed record of all of the activities in which each individual participates; we do not know, for instance, when an individual is sleeping, eating, walking, or even browsing the web. The resulting uncertainty in deliberate human activity thus poses a fundamental challenge to quantifying and modeling human behavior. Researchers commonly account for uncertainty or lack of information through stochastic models. One of the simplest stochastic models for human activity is a point process in which independent events occur at a constant rate ρ. Such processes are referred to as homogeneous Poisson processes, and they are used to describe a large class of phenomena, including some aspects of human activity [22] . Homogeneous Poisson processes have two well-known statistical properties: the time between consecutive events, the interevent time τ , follows an exponential distribution, p(τ ) = ρe −ρτ , and the number of events NT during a time interval of duration T time units follows a Poisson distribution with mean ρT . Several recent studies of deliberate human activity, including email correspondence, have focused on the former property. These studies have reported that the empirical distribution of inter-event times decays asymptotically as a power-law, p(τ ) ∝ τ −α , with exponent α ≈ 1 [23, 13, 14, 18] . Other studies have identified a similar power-law scaling in the inter-event time distribution of many other facets of human behavior, such as file downloads [10, 11, 12] [17, 19] , human locomotor activity [20] , and telephone communication [21] . These observations are in stark contrast with the predictions of a homogeneous Poisson process, suggesting that a more suitable null model with which to compare mechanistic models of human activity is a truncated power-law model with scaling exponent α = 1. * The heavy-tailed nature of the distribution of inter-event times prompts us to search for the mechanisms responsible for its emergence. Two main classes of mechanisms can be considered: (i) human behavior is primarily driven by rational decision making, which introduces correlations in activity thereby giving rise to heavy-tails; † (ii) human behavior is primarily driven by external factors such as circadian and weekly cycles, which introduces a set of distinct characteristic time scales thereby giving rise to heavy-tails. ‡ While the former interpretation has been shown to give rise to a truncated powerlaw distribution of inter-event times, the latter has been rejected by some authors [17] . Indeed, even though Hidalgo [24] investigated a model with seasonal changes in activity rates that is able to generate data with an approximate power-law decay in the distribution of inter-event times with exponents α ≈ 2 or α ≈ 1, the α ≈ 1 case requires a specific relationship between the rates of activity ρi and the corresponding duration of the seasons Ti over which each rate holds. It has therefore been argued that seasonality alone can only robustly give rise to heavy-tailed inter-event time distributions with exponent α ≈ 2 [17] . Here, we demonstrate that the distribution of inter-event times in e-mail correspondence patterns display systematic deviations from the truncated power-law null model due to circadian and weekly pat- § To whom correspondence should be addressed. E-mail: amaral@northwestern.edu c 2007 by The National Academy of Sciences of the USA * For simplicity we use a truncated power-law with an exponent of α = 1 as our null model. Similar conclusions are reached when the power-law scaling exponent is fit to the data or when other heavy-tailed null models (e.g. log-normal or log-uniform distributions [16] ) are considered. † If humans make decisions based on their own previous memories, then we might expect that humans are heavily influenced by recent events. That is, the probability ρ dt that an event will happen in a time interval dt is not constant, but is instead a decreasing function of the time elapsed since the last event [18] . ‡ This interpretation does not rely on highly competent human behavior and allows for the possibility that human activity, and hence the time dependence of ρ, is modulated by instinct, the environment, or social stimuli. www.pnas.org --PNAS Issue Date Volume Issue Number 1-9
For instance, REF proposed a nonhomogeneous PP to explain the interevent times distribution.
A Poissonian explanation for heavy-tails in e-mail communication
PNAS 105(47): 18153-18158 (2008)
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