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 analyze empirical event sequences from different domains to show that the bursty trains in these processes are hierarchically structured across different timescales in such a way that the number of bursty trains at a shorter timescale that make up a single bursty train at a longer timescale follows a power-law distribution. We propose a dynamic algorithm in which dynamics at different timescales are controlled by independent memory processes and demonstrate that it generates event sequences showing empirically observed hierarchical structures.