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.

Figure 1

An example of bursty trains detected using a time window $\mathrm{\Delta }t$ for an event sequence in time. Each vertical line denotes an event, and the numbers indicate the sizes of bursty trains.

Figure 2

The analytical solution of $M$ in Eq. (19) as a function of $\beta$ in Eq. (20) for several values of $\alpha$ in Eq. (21) (solid lines), compared with corresponding numerical results (symbols with error bars). In panel (a) we use the pure power-law distribution of $P\left(\tau \right)$ in Eq. (21), with infinite exponential cutoff, i.e., ${\tau }_c\to \infty$, while the general form of $P\left(\tau \right)$ with ${\tau }_c={10}^3{\tau }_{\min }$ is used in panel (b). The inset shows the same result as in panel (b), but in a semilog scale. Each point and its standard deviation are obtained from 50 event sequences of size $n=5\times {10}^5$.

Figure 3

(a) The analytical solution of $M$ in Eq. (35) as a function of $\beta$ in Eq. (58) for several values of $\alpha$ in Eq. (21) and $s=4$ in Eq. (59) (solid lines), compared with corresponding numerical results (symbols with error bars). We use the pure power-law distribution of $P\left(\tau \right)$ in Eq. (21), with infinite exponential cutoff, i.e., ${\tau }_c\to \infty$. (b) We show the dependence of the analytical solution of $M$ on the parameter $s$ for a fixed value of $\alpha =3.5$. (c) The same as (a) but for the general form of $P\left(\tau \right)$ with ${\tau }_c={10}^3{\tau }_{\min }$. In panels (a) and (c), each point and its standard deviation are obtained from 50 event sequences of size $n=5\times {10}^5$.