Measuring burstiness for finite event sequences

Characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand underlying mechanisms behind such complex systems and, hence, even to predict and control them. Temporal inhomogeneities in event sequences have been described in terms of bursts that are rapidly occurring events in short time periods alternating with long inactive periods. The bursts can be quantified by a simple measure, called the burstiness parameter, which was introduced by Goh and Barabási [Europhys. Lett. 81, 48002 (2008)]. The burstiness parameter has been widely used due to its simplicity, which, however, turns out to be strongly affected by the finite number of events in the time series. As the finite-size effects on burstiness parameter have been largely ignored, we analytically investigate the finite-size effects of the burstiness parameter. Then we suggest an alternative definition of burstiness that is free from finite-size effects and yet simple. Using our alternative burstiness measure, one can distinguish the finite-size effects from the intrinsic bursty properties in the time series. We also demonstrate the advantages of our burstiness measure by analyzing empirical data sets.

Figure 1

Schematic diagram of the localized model: $n$ events are localized in the period $\mathrm{\Delta }$ beginning at $t_0$ in $[0,T)$, and they are separated from each other at least by ${\tau }_0$.

Figure 2

(a) Analytic results of $B_n(x,y)$ for three reference cases: Equation (18) for regular time series, Eq. (19) for random time series, and Eq. (20) for the bursty limit. Numerical results for the random case are plotted for comparison to the analytic results. (b) Comparison of the alternative burstiness measure $A_n\left(r\right)$ in Eq. (22) to the original burstiness parameter $B\left(r\right)$ in Eq. (1) for several values of $n$. (c) Scatter plot of $B\left(r\right)$ for individual Twitter users. (d) The same as (b) but using the Twitter data set.

Figure 3

(a) Analytic results of $B_n(x,\tilde{y})$ for bursty limit and random cases and that of $B_n\left[r_n^{*}\left(\tilde{y}\right)\right]$ for the regular case when $n=10$. (b) Comparison of the alternative burstiness measure $A_{n,\tilde{y}}\left(r\right)$ in Eq. (28) to the original burstiness parameter $B\left(r\right)$ in Eq. (1) for several values of $\tilde{y}$ for $n=10$. Each gray dot in both panels corresponds to each individual Twitter user with $n=10$.