Contextual analysis framework for bursty dynamics

To understand the origin of bursty dynamics in natural and social processes we provide a general analysis framework in which the temporal process is decomposed into subprocesses and then the bursts in subprocesses, called contextual bursts, are combined to collective bursts in the original process. For the combination of subprocesses, it is required to consider the distribution of different contexts over the original process. Based on minimal assumptions for interevent time statistics, we present a theoretical analysis for the relationship between contextual and collective interevent time distributions. Our analysis framework helps to exploit contextual information available in decomposable bursty dynamics.

Figure 1

An example of event series with various contexts A, B, and C, presented in (a) a real time frame and in (b) an ordinal time frame. $l$ and $\tau$ represent the collective and contextual real interevent times, respectively. $n$ represents the contextual ordinal interevent time, while every collective ordinal interevent time is trivially 1.

Figure 2

Phase diagram summarizing the relation between heavy-tailed distributions of $l$, $\tau$, and $n$, in terms of their corresponding power-law exponents $\alpha$, ${\alpha }^{\prime }$, and $\beta$. Contextual bursts in a real time frame are dominated by contextual bursts in an ordinal time frame if $\alpha >2$ and $\alpha >\beta$, by collective bursts if $\beta >2$ and $\beta >\alpha$, or otherwise characterized by both kinds of bursts.

Figure 3

Numerical results of the model for $\alpha =\frac{3}{2}$ (top), $\frac{5}{2}$ (middle), and 4 (bottom), all with $l_0=1$. (a) Numerical results of $F_n\left(\tau \right)$ for different values of $n$ collapsed into one curve, i.e., $g_1^{-}\left(x\right)$ for $x<x_{\times }$ and $g_1^{+}\left(x\right)$ for $x>x_{\times }$, drawn with a black curve. (b) The power-law exponent ${\alpha }^{\prime }$ is estimated from $P\left(\tau \right)\sim {\tau }^{-{\alpha }^{\prime }}$ for each $\beta$ to be compared with the scaling relation ${\alpha }^{\prime }=\min \{\frac{\beta +1}{2},\frac{3}{2}\}$, denoted by a black line in the inset. (c)–(f) Scaling collapse of $F_n\left(\tau \right)$ and the estimated values of ${\alpha }^{\prime }$ comparable to ${\alpha }^{\prime }=\min \{\alpha ,\beta \}$ support our analysis for $\alpha >2$. Here ${\tau }_c$ for each $n$ is determined to maximize $F_n\left(\tau \right)$. In (c) $x_0\approx 0.659$ is used. In the inset of (e), tail parts are collapsed by $F_n\left(\tau \right)n$ vs $(\tau -{\tau }_c)n^{-1/2}$.