Modeling correlated bursts by the bursty-get-burstier mechanism

Burst-tree structure and higher-order temporal correlations

Understanding the characteristics of temporal correlations in a time series is crucial for developing accurate models in natural and social sciences. The burst-tree decomposition method was recently introduced to reveal temporal correlations in a time series in the 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)10.1038/s41598-020-68157-1]. Such structure cannot be solely captured by the interevent time distribution but can show higher-order correlations beyond interevent times. It 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 a time series.

Temporal scaling theory for bursty time series with clusters of arbitrarily many events

Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.

Individual-driven versus interaction-driven burstiness in human dynamics: The case of Wikipedia edit history

The origin of non-Poissonian or bursty temporal patterns observed in various data sets for human social dynamics has been extensively studied, yet its understanding still remains incomplete. Considering the fact that humans are social beings, a fundamental question arises: Is the bursty human dynamics dominated by individual characteristics or by interaction between individuals? In this paper we address this question by analyzing the Wikipedia edit history to see how spontaneous individual editors are in initiating bursty periods of editing, i.e., individual-driven burstiness, and to what extent such editors' behaviors are driven by interaction with other editors in those periods, i.e., interaction-driven burstiness. We quantify the degree of initiative (DoI) of an editor of interest in each Wikipedia article by using the statistics of bursty periods containing the editor's edits. The integrated value of the DoI over all relevant timescales reveals which is dominant between individual-driven and interaction-driven burstiness. We empirically find that this value tends to be larger for weaker temporal correlations in the editor's editing behavior and/or stronger editorial correlations. These empirical findings are successfully confirmed by deriving an analytic form of the DoI from a model capturing the essential features of the edit sequence. Thus our approach provides a deeper insight into the origin and underlying mechanisms of bursts in human social dynamics.

Burst-tree decomposition of time series reveals the structure of temporal correlations

Comprehensive characterization of non-Poissonian, bursty temporal patterns observed in various natural and social processes is crucial for understanding the underlying mechanisms behind such temporal patterns. Among them bursty event sequences have been studied mostly in terms of interevent times (IETs), while the higher-order correlation structure between IETs has gained very little attention due to the lack of a proper characterization method. In this paper we propose a method of representing an event sequence by a burst tree, which is then decomposed into a set of IETs and an ordinal burst tree. The ordinal burst tree exactly captures the structure of temporal correlations that is entirely missing in the analysis of IET distributions. We apply this burst-tree decomposition method to various datasets and analyze the structure of the revealed burst trees. In particular, we observe that event sequences show similar burst-tree structure, such as heavy-tailed burst-size distributions, despite of very different IET distributions. This clearly shows that the IET distributions and the burst-tree structures can be separable. The burst trees allow us to directly characterize the preferential and assortative mixing structure of bursts responsible for the higher-order temporal correlations. We also show how to use the decomposition method for the systematic investigation of such correlations captured by the burst trees in the framework of randomized reference models. Finally, we devise a simple kernel-based model for generating event sequences showing appropriate higher-order temporal correlations. Our method is a tool to make the otherwise overwhelming analysis of higher-order correlations in bursty time series tractable by turning it into the analysis of a tree structure.

Copula-based algorithm for generating bursty time series

Dynamical processes in various natural and social phenomena have been described by a series of events or event sequences showing non-Poissonian, bursty temporal patterns. Temporal correlations in such bursty time series can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. Modeling and simulating various dynamical processes requires us to generate event sequences with a heavy-tailed IET distribution and memory effects between IETs. For this, we propose a Farlie-Gumbel-Morgenstern copula-based algorithm for generating event sequences with correlated IETs when the IET distribution and the memory coefficient between two consecutive IETs are given. We successfully apply our algorithm to the cases with heavy-tailed IET distributions. We also compare our algorithm to the existing shuffling method to find that our algorithm outperforms the shuffling method for some cases. Our copula-based algorithm is expected to be used for more realistic modeling of various dynamical processes.

Analytically solvable autocorrelation function for weakly correlated interevent times

Long-term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions. Such temporal correlations can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. In contrast to the role of heterogeneous IETs on the autocorrelation function, little is known about the effects due to the correlations between IETs. To rigorously study these effects, we derive an analytical form of the autocorrelation function for the arbitrary IET distribution in the case with weakly correlated IETs, where the Farlie-Gumbel-Morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive IETs. Our analytical results are confirmed by numerical simulations for exponential and power-law IET distributions. For the power-law case, we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between IETs. Our analytical approach enables us to better understand long-term temporal correlations induced by the correlations between IETs.

Correlated bursts in temporal networks slow down spreading

Spreading dynamics has been considered to take place in temporal networks, where temporal interaction patterns between nodes show non-Poissonian bursty nature. The effects of inhomogeneous interevent times (IETs) on the spreading have been extensively studied in recent years, yet little is known about the effects of correlations between IETs on the spreading. In order to investigate those effects, we study two-step deterministic susceptible-infected (SI) and probabilistic SI dynamics when the interaction patterns are modeled by inhomogeneous and correlated IETs, i.e., correlated bursts. By analyzing the transmission time statistics in a single-link setup and by simulating the spreading in Bethe lattices and random graphs, we conclude that the positive correlation between IETs slows down the spreading. We also argue that the shortest transmission time from one infected node to its susceptible neighbors can successfully explain our numerical results.

Hierarchical burst model for complex bursty dynamics

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 γ, the interevent time distribution with a power-law exponent α, 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. 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 multilevel causal or decision-making process. By studying our model analytically and numerically, we confirm the scaling relation α+γ=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.

Limits of the memory coefficient in measuring correlated bursts

Temporal inhomogeneities in event sequences of natural and social phenomena have been characterized in terms of interevent times and correlations between interevent times. The inhomogeneities of interevent times have been extensively studied, while the correlations between interevent times, often called correlated bursts, are far from being fully understood. For measuring the correlated bursts, two relevant approaches were suggested, i.e., memory coefficient and burst size distribution. Here a burst size denotes the number of events in a bursty train detected for a given time window. Empirical analyses have revealed that the larger memory coefficient tends to be associated with the heavier tail of the burst size distribution. In particular, empirical findings in human activities appear inconsistent, such that the memory coefficient is close to 0, while burst size distributions follow a power law. In order to comprehend these observations, by assuming the conditional independence between consecutive interevent times, we derive the analytical form of the memory coefficient as a function of parameters describing interevent time and burst size distributions. Our analytical result can explain the general tendency of the larger memory coefficient being associated with the heavier tail of burst size distribution. We also find that the apparently inconsistent observations in human activities are compatible with each other, indicating that the memory coefficient has limits to measure the correlated bursts.