Burst-tree structure and higher-order temporal correlations

TL;DR

This paper investigates how different burst-merging kernels influence higher-order temporal correlations in time series, revealing their effects on burst size distributions, memory, and autocorrelation functions, with analytical insights for some kernels.

Contribution

It introduces a detailed analysis of how various burst-merging kernels affect temporal correlations, including analytical solutions for certain kernels, advancing understanding of bursty train structures.

Findings

Kernels with preferential merging produce heavy-tailed burst size distributions.

Kernels with assortative merging create positive correlations between burst sizes.

The autocorrelation decay depends on the kernel and the interevent time distribution's power-law exponent.

Abstract

Understanding characteristics of temporal correlations in time series is crucial for developing accurate models in natural and social sciences. The burst-tree decomposition method was recently introduced to reveal higher-order temporal correlations in time series in a form of an event sequence, in particular, the hierarchical structure of bursty trains of events for the entire range of timescales [Jo et al., Sci.~Rep.~\textbf{10}, 12202 (2020)]. Such structure has been found to be simply characterized by the burst-merging kernel governing which bursts are merged together as the timescale for defining bursts increases. In this work, we study the effects of kernels on the higher-order temporal correlations in terms of burst size distributions, memory coefficients for bursts, and the autocorrelation function. We employ several kernels, including the constant, sum, product, and diagonal kernels as well as those inspired by empirical results. We generically find that kernels with preferential merging lead to heavy-tailed burst size distributions, while kernels with assortative merging lead to positive correlations between burst sizes. The decaying exponent of the autocorrelation function depends not only on the kernel but also on the power-law exponent of the interevent time distribution. In addition, thanks to the analogy to the coagulation process, analytical solutions of burst size distributions for some kernels could be obtained. Our findings may shed light on the role of burst-merging kernels as underlying mechanisms of higher-order temporal correlations in time series.