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 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.
Highlights
Events in temporal patterns of natural and social phenomena have often been found to be inhomogeneously distributed in time
To understand the hierarchical temporal structure, we devise a hierarchical burst model by assuming that each observed event in an event sequence might be a consequence of the multilevel causal or decision-making process: A seed event at the zeroth level induces other events at the first level, each of which in turn leads to other events at the second level, and so on
We introduce the hierarchical burst model by assuming that each observed event in an event sequence might be a consequence of the multilevel causal or decision-making process: A seed event at the zeroth level induces other events at the first level, each of which in turn leads to other events at the second level, and so on
Summary
Events in temporal patterns of natural and social phenomena have often been found to be inhomogeneously distributed in time. Examples include solar flares [1], earthquakes [2,3], neuronal firing [4], and human social activities [5,6] Such temporal inhomogeneities in event sequences have been studied in terms of bursts, which are rapidly occurring events in short-time periods, alternating with long periods of inactivity. Thanks to the simplicity of our model, we can derive a fractal dimension df of the event sequence, the decaying exponent γ of the autocorrelation function, and the power-law exponent α of the interevent time distribution to confirm the scaling relation α + γ = 2, while the derivation of the burst size distributions turns out to be not straightforward. Our paper is organized as follows: In Sec. II, after introducing the hierarchical burst model, we study our model analytically and numerically in terms of the scaling behaviors of the fractal temporal structure, the autocorrelation function, and the interevent time distribution.
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