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

It is not straightforward to devise a model or process that can answer this question, because the heavy tail of the burst size distribution typically implies that arbitrarily many consecutive interevent times may be correlated with each other. It is worth noting that correlations between two consecutive interevent times have been quantified in terms of the memory coefficient,⁴⁹ local variation,⁵⁰ and mutual information;⁵¹ they were implemented using, e.g., a copula method^52,53 and a correlated Laplace Gillespie algorithm in the context of many-body systems.^54,55 Correlations between an arbitrary number of consecutive interevent times have been modeled by means of, e.g., the two-state Markov chain,⁴² self-exciting point processes,⁵⁶ the interevent time permutation method,⁵⁷ and a model inspired by the burst-tree decomposition method.⁴⁸ Although scaling behaviors of the autocorrelation function were studied in some of mentioned works,^42,53,56,57 the scaling relation $γ(α,β)$ has not been clearly understood due to the lack of analytical solutions of the autocorrelation function.

In this work, we devise a model for generating a time series with arbitrary functional forms of the interevent time distribution and burst size distribution. Assuming power-law tails for interevent time and burst size distributions, our model generates correlations between an arbitrary number of consecutive interevent times. By theoretically analyzing the model, we derive asymptotically exact solutions of the autocorrelation function from the model time series, thereby enabling us to find the scaling relation $γ(α,β)$⁠. These analytical results will help us to better understand temporal scaling behaviors in empirical time series.

The paper is organized as follows. In Sec. II, we introduce the model with arbitrary functional forms of interevent time and burst size distributions. In Sec. III, we provide an analytical framework for the derivation of the autocorrelation function for the model time series. In Sec. IV, by assuming power-law distributions of interevent times and burst sizes, we derive analytical solutions of the autocorrelation function, hence the decay exponent $γ$ as a function of $α$ and $β$⁠. We also compare the obtained analytical results with numerical simulations. Finally, we conclude our paper in Sec. V.

Let us introduce the following model for generating event sequences ${x(1),…,x(T)}$ in discrete time, where $T$ is the number of discrete times, for given interevent time distribution and burst size distribution. By definition, $x(t)=1$ if an event occurs at time $t∈{1,…,T}$⁠, and $x(t)=0$ otherwise. By construction, the minimum interevent time is $τ min=1$⁠. Furthermore, we only consider bursts with $Δt=1$ throughout this work. In other words, we regard that events occurring in consecutive times form a burst, including the case in which the burst contains only one event.

The event sequence $x(t)$ is generated using an interevent time distribution for $τ≥2$⁠, denoted by $ψ(τ)$⁠, and a burst size distribution $Q(b)$⁠. Note that $ψ(τ)$ is normalized. To generate an event sequence, we first randomly draw a burst size $b 1$ from $Q(b)$ to set $x(t)=1$ for $t=1,…, b 1$⁠. Then, we randomly draw an interevent time $τ 1$ from $ψ(τ)$ to set $x(t)=0$ for $t= b 1+1,…, b 1+ τ 1−1$⁠. Note that $τ 1≥2$⁠. We draw another burst size $b 2$ from $Q(b)$ and another interevent time $τ 2$ from $ψ(τ)$⁠, respectively, to set $x(t)=1$ for $t= b 1+ τ 1,…, b 1+ τ 1+ b 2−1$ and $x(t)=0$ for $t= b 1+ τ 1+ b 2,…, b 1+ τ 1+ b 2+ τ 2−1$⁠. We repeat this procedure until $t$ reaches $T$⁠. See Fig. 1(a) for an example.

Let us consider the case in which $x(t)x(t+ t d)$ is non-zero, i.e., $x(t)=1$ and $x(t+ t d)=1$⁠. As depicted in Fig. 1(a), the time series $x(t)$ in a period of $[t,t+ t d]$ is typically composed of several alternating bursts and interevent times larger than one. Here, the consecutive interevent times of $τ=1$ forms a burst and the sum of such interevent times of $τ=1$ is equal to the burst size minus one. Therefore, the time lag $t d$ is written as a sum of burst sizes (each minus one) of bursts appeared in the period of $[t,t+ t d]$ and interevent times between those bursts. We note that the first burst contains an event at time $t$ and that the last burst contains an event at time $t+ t d$⁠.

We now divide the entire range of $α,β>1$ into several cases to derive the analytical solution of the autocorrelation function in each case. Considering the symmetric nature of $γ$ in Eq. (39), the following cases are sufficient to get the complete picture of the result.