Hierarchical organization of bursty trains in event sequences

TL;DR

The paper addresses the prevalence of bursty dynamics in diverse real-world systems and shows that bursts form a hierarchical structure across multiple timescales, driving higher-order temporal correlations beyond interevent-time distributions. It introduces a rigorous framework for bursts, mergeings across scales, and burst trees, supported by empirical analysis of neuronal, social, and seismic data. A dynamic generative algorithm with multilevel memory is developed, capable of reproducing heavy-tailed burst sizes and merging distributions across scales, and is demonstrated across various timescale sequences. The work suggests that memory mechanisms operate multi-dimensionally over time and offers a flexible, canonical model for generating synthetic event sequences with realistic hierarchical structure and correlations. This has practical implications for understanding and simulating complex temporal processes in natural and social systems.

Abstract

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.

Figures (4)

• Figure 1: Schematic diagram for the hierarchical structure of bursty trains at multiple timescales. Each vertical line in the time axis represents an event. $b_{\delta,j}$ denotes the size of the $j$th burst in the burst size sequence at the timescale $\delta$. Each $b_{\delta_l,j}$ is the sum of multiple consecutive $b_{\delta_{l-1},j'}$s, the number of which is called a merging number and denoted by $k_{j}$. We have assumed that $\delta_0<\tau_{\rm min}\le \delta_1$, with $\tau_{\rm min}$ denoting the minimum interevent time (IET), leading to $b_{\delta_0,j}=1$, hence $b_{\delta_1,j}=k_{j}$ from Eq. \ref{['eq:k_define']}. The number $l$ below each IET in the time axis denotes the level determining the range of the corresponding IET.
• Figure 2: Summary of event sequence analysis of empirical data: spikes of a rat neuron (top), edits made by a Wikipedia editor (middle), and earthquakes in Southern California (bottom). For each colored section, the top row shows the distributions of interevent times (left), burst sizes (center), and merging numbers (right). For the burst size and merging number distributions, the maximum likelihood estimators of the power law exponents are shown at the bottom. Fitted exponents are indicated by filled circles if the likelihood ratio test supports a power law distribution stronger than an exponential distribution with significance $p < 0.05$, and by open circles otherwise. The standard errors of the estimated exponents are smaller than the symbols and therefore not shown. The values of $\delta$ and $\delta'$ span the entire range of relevant timescales.
• Figure 3: Illustrative example of the incremental dynamic algorithm. Colored circles indicate the state of the system at each iteration, with the number on top of each circle denoting the value of $\kappa_l$. Rightward horizontal arrows represent the system sampling an IET and generating an event, after which it moves down to the lowest level. Upward vertical arrows represent transitions to the level above. The spike series at the bottom shows the generated event sequence.
• Figure 4: Summary of event sequence analysis of synthetic data generated by the incremental dynamic algorithm with linear timescales (top), quadratic timescales (middle), and exponential timescales (bottom). Statistical analyses, panel arrangement, and symbol conventions are the same as in Fig. \ref{['fig:empirical']}.