Hierarchical burst model for complex bursty dynamics

TL;DR

This paper introduces a hierarchical burst model to explain complex bursty dynamics in natural and social phenomena, revealing scaling relations and correlations in temporal event sequences through analytical and numerical analysis.

Contribution

The paper proposes a novel hierarchical burst model that captures multi-level causal processes and explains observed temporal scaling behaviors and correlations in bursty event sequences.

Findings

Confirmed the scaling relation α + γ = 2 despite correlations.

Supported the existence of stretched exponential burst size distributions.

Observed log-periodic behavior in the autocorrelation function.

Abstract

Temporal inhomogeneities observed in various natural and social phenomena have often been characterized in terms of scaling behaviors in the autocorrelation function with a decaying exponent $\gamma$, the interevent time distribution with a power-law exponent $\alpha$, and the burst size distributions. Here the interevent time is defined as a time interval between two consecutive events in the event sequence, and the burst size denotes the number of events in a bursty train detected for a given time window. In order to understand such temporal scaling behaviors implying a hierarchical temporal structure, we devise a hierarchical burst model by assuming that each observed event might be a consequence of the multi-level causal or decision-making process. By studying our model analytically and numerically, we confirm the scaling relation $\alpha+\gamma=2$, established for the uncorrelated interevent times, despite of the existence of correlations between interevent times. Such correlations between interevent times are supported by the stretched exponential burst size distributions, for which we provide an analytic argument. In addition, by imposing conditions for the ordering of events, we observe an additional feature of log-periodic behavior in the autocorrelation function. Our modeling approach for the hierarchical temporal structure can help us better understand the underlying mechanisms behind complex bursty dynamics showing temporal scaling behaviors.