A Poissonian explanation for heavy tails in e-mail communication

By R. Dean Malmgren, Daniel B. Stouffer, Adilson E. Motter, Luís A. Nunes Amaral. 2008

422 citations recorded by the source corpus.

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 non-Poissonian. Here, we demonstrate that the approximate power-law scaling of the inter-event time distribution is a consequence of circadian and weekly cycles of human activity. We propose a cascading nonhomogeneous Poisson process that 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.

Subfields: Molecular Biology, Statistical and Nonlinear Physics

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Citations in the corpus

Beyond Word Frequency: Bursts, Lulls, and Scaling in the Temporal Distributions of Words (2009, background): “[5] Malmgren RD, Stouffer DB, Motter AE, Amaral LAN”
Patterns of temporal variation in online media (2011, background): “Such content exhibits rich temporal dynamics [21, 22, 26] and is a direct reflection of the attention that people pay to various topics.”
Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter (2011, background): “There has also been work on the temporal patterns of information diffusion — the rate over time at which different pieces of information are adopted [9, 18, 21, 24, 30].”
Universal features of correlated bursty behaviour (2011, background): “[27] Malmgren RD, Stouffer DB, Motter AE, Amarala LAN (2008) A Poissonian explanation for heavy tails in e-mail communication.”
Understanding Short-term Changes in Online Activity Sessions (2017, background): “Moreover, it has been shown that people tend to reply to emails faster during the mornings and in weekdays [13], that email activity is bursty yet predictable [16] and that even postal letter replies exhibit certain regularities [17].”
Recurrent Marked Temporal Point Processes: Embedding Event History to Vector (2016, methodology): “In human activity modeling, Poisson Process and its variants, have been used to model the inter-event durations of human activities [29, 15].”
Do the Rich Get Richer? An Empirical Analysis of the Bitcoin Transaction Network (2013, result): “9), in agreement with the behavior widely observed in various complex systems [31, 35–37].”
The TimeGeo modeling framework for urban mobility without travel surveys (2016, neutral): “Compared with previous models that randomly draw the stay duration (or waiting time Δt) or the number of visited locations (N) from aggregated empirical distributions (14, 15, 29), by introducing three individual-specific parameters including weekly home-based tour number nw, dwell rate β1, and burs”
Origins of Homophily in an Evolving Social Network1 (2009, background): “Although one could study the dynamics of these spike trains as a phenomenon in itself (e.g., Barabási 2005; Malmgren et al. 2008), here we treat them simply as the observable signature of an unobservable social network that is persistent and continuous and is also evolving in time.”
Meme-tracking and the dynamics of the news cycle (2009, background): “One could take this decrease to be exponential in some polynomial function of t − tj based on research on novelty and attention [32]; or, following research on human response-time dynamics [24, 29], one could take it to be a heavy-tailed functional form.”
Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions (2012, methodology): “This is the procedure used by Clauset et al. (2009) and Malmgren et al. (2008), but it is not clear if it is the one of Goldstein et al. (2004).”