Burst-tree structure and higher-order temporal correlations

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.~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 detecting 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, additive, and product kernels as well as those inspired by the empirical results. We find that kernels with preferential mixing lead to the heavy-tailed burst size distributions, while kernels with assortative mixing 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.