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 $\gamma$, the interevent time distribution with a power-law exponent $\alpha$, 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 $\alpha +\gamma =2$, 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.

Figure 1

(a) Schematic diagram of the hierarchical burst model with $\eta =2$ up to the second level. Red vertical arrows indicate the events at each level, each of which is assigned an induction interval (light green shade with horizontal dotted arrow) for events at the next level. (b) An example of the event sequence generated by the model with $\eta =2, \lambda =2.5, R=1$, and $L=8$. See the text for the details of the model.

Figure 2

Simulation results of the hierarchical burst model using $\eta =2$. In (a)–(c), the box counting result $N\left(r\right)$ for measuring the fractal dimension, the autocorrelation function $A\left(t_d\right)$, and the interevent time distribution $P\left(\tau \right)$ in the case with $\lambda =2.5$ and $L=18$ (circles) are compared to the analytical results for $d_f, \gamma$, and $\alpha$ (solid lines), respectively. Each curve was averaged over 500 realizations of event sequences. In (d)–(f), for the same value of $\lambda =2.5$, we plot the fractal dimension $d_f\left(n\right)$, the decaying exponent $\gamma \left(n\right)$ of the autocorrelation function, and the power-law exponent $\alpha \left(n\right)$ of the interevent time distribution, as functions of $n={\eta }^L$, i.e., the number of events in the event sequence (circles). For fitting the data, we adopt the functional form of $f\left(n\right)=f\left(\infty \right)+a{n}^{-\nu }$ with $\nu >0$ (solid lines). In (g), we plot numerical values of $d_f, \gamma$, and $\alpha$ using various values of $\lambda$ for fixed $\eta =2$ and $L=18$, where each point (circle) was averaged over 200 realizations of event sequences, to confirm the scaling relations of $\alpha =1+d_f$ (dotted line) and $\gamma =1-d_f$ (solid line).

Figure 3

(a) Simulation results of the burst size distributions $Q_{\mathrm{\Delta }t}\left(b\right)$ for various time windows $\mathrm{\Delta }t$ by the hierarchical burst model using $\eta =2, \lambda =2.5$, and $L=18$ (symbols), fitted with a stretched exponential function (black solid line). Here ${\langle b\rangle }_{\mathrm{\Delta }t}$ is the average burst size for a given $\mathrm{\Delta }t$. (b) A schematic diagram for the analytic calculation of probability distributions of interevent time between events induced by the same (or different) events at the $(l-1)\mathrm{th}$ level, denoted by $P\left({\tau }_1\right)$ [$P\left({\tau }_2\right)]$. (c) Comparison between $Pr({\tau }_1<\mathrm{\Delta }t)$ in Eq. (19) and $Pr({\tau }_2<\mathrm{\Delta }t)$ in Eq. (29) for $\mathrm{\Delta }t=1/{\lambda }^l$.

Figure 4

Log-periodic behaviors in the autocorrelation functions of the hierarchical burst model using several values of $(\eta ,L)=(2,17)$, (3,11), and (4,9) for a fixed $\lambda =4.8$ under conditions in Eqs. (31) and (32) for nonoverlapping induction intervals, where all curves were averaged over 200 realizations of event sequences. For each curve, we have used the best fit value of ${\gamma }_{\mathrm{fit}}$ for the best presentation of the log-periodicity. The curve for $\eta =2$ was vertically shifted for the clear presentation.

Figure 5

Simulation results of the hierarchical burst model using $\eta =2, \lambda =2.5, R=0.1$, and $L=18$, with 10 seed events at the zeroth level: (a) Box counting result for measuring the fractal dimension, (b) the autocorrelation function, (c) the interevent time distribution, and (d) the burst size distributions for various values of $\mathrm{\Delta }t$. Each curve was averaged over 100 realizations of event sequences. Black solid lines in (a)–(c) represent the analytic results, while the black solid line in (d) shows a stretched exponential function fitted to the data.