In order to understand the temporal features of successful petitions with a large number of signatures, we divide all petitions into four different classes based on the number of signatures N, cf. Fig 1. In subsequent paragraphs we refer to them as signature number classes. Class 1 contains petitions with the smallest numbers of signatures and class 4 the ones with the largest numbers respectively. The numbers of petitions in each class are: 2182, 1213, 151 and 8 (ascending class index). The probability density function of the inter-event times is shown in Fig 1 (left). As expected, inter-event times are more broadly distributed than the exponential distribution expected from a Poisson process. In Fig 1 (right), we show the relative frequency of signing activity per second. Most of the signing activity corresponds to one signing event per time stamp. The time series of signing events clearly exhibits bursty behavior, i.e. there are periods of high signing activity followed by periods of low or no activity. As an example, we show the time series of the signatures per hour of the petition with the largest number of signatures in Fig 2 (left). The time evolution of the total number of signatures is illustrated in Fig 2 (right). As illustrated in the inset of Fig 2 (left), we also observe a circadian rhythm in the petitions’ time series, which is typical for human activities. However, it is clear that bursty behavior is observed on a larger time scale. We find that the heterogeneity of inter-event time distributions holds independently of the popularity of the petition, looking at the behavior of the four different classes of numbers of signatures. However, we note that there is a systematic increase in the frequency of short inter-event times and decrease in longer inter event-times as the popularity of a petition increases.

Before discussing the actual time series analysis, we shortly analyze the petitions in our data set in terms of their number of signatures. In previous studies it has been found that only a small fraction of petitions accumulate a considerable amount of signatures [2, 3]. For the petitions in our data set a Zipf plot (number of signatures vs. petition rank) is shown in Fig 3 (left). The red lines are guides to the eye with slopes −0.9 and −8.0. The change in slope has been also observed previously for another online petition platform [3]. However, in the latter case the change occurs at a number of signatures of around 500 which corresponds to the predefined quorum of 500 in their data set and led the authors to the conclusion that petitions are less eager to accrue additional signatures after meeting the number required for an official government response. In our data there is no fixed quorum value since the openPetition quorum scales with the number of inhabitants in the target region. The different slopes might be a consequence of different growth dynamics. In Fig 3 (right) we show the probability distribution of the numbers of signatures. Around 30% of all petitions in our data set only have one or zero signatures. However, some petitions (0.01%) acquired more than 100.000 signatures. The inset in Fig 3 (right) shows good agreement between the distribution of the numbers of signatures’ first digits and the expected Benford distribution which is a typical feature of logarithmically distributed data [16, 17].

To further analyze the temporal features of signing time series, we employ tools for non-stationary time series. We construct signing spike trains, i.e. the time series of discrete signing events, for each petition. Here a spike represents signing activity at the corresponding time. We are thus not taking into account multiple signing activity—a legitimate approximation since nearly all signing activity corresponds to one signing event per time stamp, cf. Fig 1 (right).

As a null model for comparison, we generate a randomized sequence of the original inter-event intervals within the considered time interval, i.e. the total number of seconds between the petition’s start and end [13]. This procedure destroys the inter-event time correlations but preserves the distribution. Using this null model allows to study if correlations are the key factor to produce the observed local variation.

Before analyzing the local variation L_V of the petition time series, we apply a statistical F-test as suggested in Refs. [15, 18] to decide whether L_V consistently characterizes the time series. This means that the variance of L_V across different periods in one time series should be smaller than the variance in the population of all time series. Here we subdivide each time series in 20 slices and calculate the corresponding F-values as the ratios between the variance of L_V in the population of all time series and the variances across the 20 slices [15, 18]. For different popularity classes, we show the F-values of L_V in Table 1. The F-values are significantly larger than the 0.1-percentile values. This suggests that the variance of L_V in a single time series is significantly smaller than the variance in the population and we conclude that the local variation L_V properly characterizes our time series.

We now compute the local variation L_V as defined in Eq (1) and see differences in the distributions of L_V between the real spike train data in Fig 4 (upper left) and the randomized null model data in Fig 4 (upper right). For the eight petitions with the largest numbers of signatures, we present the corresponding L_V values in Table 2. Popular petitions with large numbers of signatures exhibit smaller values of L_V compared to less popular ones suggesting that the former are less bursty. As described in the Materials and Methods section, we expect 〈L_V〉 = 1 for a Poisson process and we find deviations for the real data as illustrated in Fig 4 (lower left). To capture the strength of the deviation we compute the z -value = ( μ ( L V ) - μ 0 ) / σ ( L V ) / n, where μ₀ = 1, μ(L_V) and σ(L_V) define the mean and standard deviation of a given L_V distribution respectively and n is the number of data points. The z-values of the randomized spike trains are almost zero indicating a Poissonian signal whereas petitions with low numbers of signatures exhibit a very large z-value, indicating anomalously high L_V, as shown in Fig 4 (lower right). This indicates that petitions with fewer signatures deviate more from the Poissonian null model. The results for L_V are in agreement with a similar analysis involving L_V of different popularity classes in Twitter data [13, 14]. In Ref. [13], popular hashtags were found to exhibit less bursty spike trains compared to less popular ones and in Ref. [14] more popular user’s activity is less bursty than that of sporadic users. It is worth noting that petition signing seems to more closely resemble the second case of user’s activity, since the popular spike trains exhibit L_V values just below 1, while in the first study L_V ≪ 1. Interestingly, when we partition petitions by their duration, i.e. the time period in which people have the possibility to sign a petition, we do not find that the burstiness varies according to the duration class cf. Fig 5.

In accordance with Ref. [13], we study the persistence of L_V through time by dividing each time series into two halves to calculate the local variation L_V(t₁) in the first half and L_V(t₂) in the second half. We find that higher numbers of signatures lead to higher correlations [18] between these two values of L_V (see Fig 6). In the past similar patterns have been observed in Twitter data [13], with the critical difference that high correlation coefficients have only been found for intermediate classes. This effect might be an artifact due to the small number of samples in classes with extremely high activity. In fact, when excluding the classes of extremely high activity the data presented in Fig 6 and the findings on twitter data [13] indicate that the distribution of L_V is getting narrower with increasing activity causing an increase in correlation.

According to the results presented in Fig 4 (upper left) that are based on the local variation L_V, popular petitions with a large number of signatures are less bursty compared to the ones with less signatures. However, a characterization based on the burstiness coefficient B as defined in Eq (3) indicates more bursty signals for larger numbers of signatures (see inset in Fig 7 (left)). Why do high signing petitions exhibit burstiness according to the burstiness coefficient B but not according to the local variation L_V? This difference is due to the an important difference in the conceptual and formal definitions of burstiness according to L_V and B. Namely, L_V takes the order of inter-event times into account, whereas B does not.

The discrepancy between the effect can be partly understood measuring the memory M of the inter-event time distribution, according to Eq (4). For the different petition classes we illustrate their burstiness coefficient, memory coefficient, and the local variation in Fig 8. For the eight petitions with the largest numbers of signatures, we present the corresponding B, M and L_V values in Table 2. The local variation L_V is based on the differences of consecutive inter-event times. However, for the computation of B the order of inter-event times does not matter. Memory is a measure of the correlation between adjacent inter-event intervals. The probability density function of M is illustrated separately in Fig 7 (right). We find a positive memory coefficient with an average value of M ≈ 0.3 as also contained in Fig 8 (right). This suggests that such correlations between adjacent intervals are an important source of burstiness in the petition’s signing process and thus B alone is not a good measure of the burstiness, as suggested in [10]. Interestingly, unlike prior analysis of patterns of human activity [10] (e.g. e-mail communication [19]), we find a non-negligible positive memory coefficient. To summarize, burstiness is a consequence of strong correlations in daily human activity and thus cannot be captured by the burstiness coefficient B alone. The memory coefficient M, as defined in Eq (4), captures some of these correlations but capture differences between subsequent inter-event times. However, the local variation L_V quantifies this local time order.