Hierarchical organization of bursty trains in event sequences

Temporal sequences of discrete events that describe natural and social processes are often driven by non-Poisson dynamics. In addition to a heavy-tailed interevent time distribution, which primarily captures the deviation from a Poisson process, a heavy tail in the distribution of bursty train sizes is frequently observed, which implies the presence of higher-order temporal correlations that extend beyond interevent times. Here, we study empirical event sequences from different domains to show that the bursty trains in these processes are hierarchically structured across different timescales, and that such hierarchical organization gives rise to the higher-order temporal correlations. We propose a dynamic algorithm that generates event sequences with hierarchical structures with arbitrary precision. The algorithm successfully reproduces the features of real-world phenomena, implying the presence of memory mechanisms embedded in system dynamics across multiple timescales.