Two-state Markov-chain Poisson nature of individual cellphone call statistics

Abstract

Unfolding the burst patterns in human activities and social interactions is a very important issue especially for understanding the spreading of disease and information and the formation of groups and organizations. Here, we conduct an in-depth study of the temporal patterns of cellphone conversation activities of 73 339 anonymous cellphone users, whose inter-call durations are Weibull distributed. We find that the individual call events exhibit a pattern of bursts, that high activity periods are alternated with low activity periods. In both periods, the number of calls are exponentially distributed for individuals, but power-law distributed for the population. Together with the exponential distributions of inter-call durations within bursts and of the intervals between consecutive bursts, we demonstrate that the individual call activities are driven by two independent Poisson processes, which can be combined within a minimal model in terms of a two-state first-order Markov chain, giving significant fits for nearly half of the individuals. By measuring directly the distributions of call rates across the population, which exhibit power-law tails, we purport the existence of power-law distributions, via the ‘superposition of distributions’ mechanism. Our findings shed light on the origins of bursty patterns in other human activities.