Temporal dynamics of online petitions

Abstract

Online petitions are an important avenue for direct political action, yet the dynamics that determine when a petition will be successful are not well understood. Here we analyze the temporal characteristics of online-petition signing behavior in order to identify systematic differences between popular petitions, which receive a high volume of signatures, and unpopular ones. We find that, in line with other temporal characterizations of human activity, the signing process is typically non-Poissonian and non-homogeneous in time. However, this process exhibits anomalously high memory for human activity, possibly indicating that synchronized external influence or contagion play and important role. More interestingly, we find clear differences in the characteristics of the inter-event time distributions depending on the total number of signatures that petitions receive, independently of the total duration of the petitions. Specifically, popular petitions that attract a large volume of signatures exhibit more variance in the distribution of inter-event times than unpopular petitions with only a few signatures, which could be considered an indication that the former are more bursty. However, petitions with large signature volume are less bursty according to measures that consider the time ordering of inter-event times. Our results, therefore, emphasize the importance of accounting for time ordering to characterize human activity.

Fig 1. Inter-event time distribution and burstiness.

All petitions are divided into four different classes based on the number of signatures N. (left) The corresponding probability density function (PDF) of the inter-event time intervals (hour). (right) The relative frequency of signing activity per second. The vast majority of signing activity corresponds to one signing event per time stamp.

Fig 2. Signing time series and time evolution of total number of signatures.

(left) Time series of the largest petition’s signing activity per hour. The inset shows the superimposed circadian pattern. (right) The corresponding total number of signatures as a function of time.

Fig 3. Characterizing the distribution of the number of signatures a petition accrues.

(left) Number of signatures as function of their rank (Zipf plot) in the openPetition data set. The red lines are guides to the eye with slopes −0.9 and −8.0 respectively. (right) Relative frequency of petitions in the openPetition data set with a certain number of signatures. The inset shows the distribution of the petitions’ signatures first digit (green bars) and the corresponding Benford distribution data (red dots).

Fig 4. Local variation analysis of petition signing spike trains for different classes of numbers of signatures.

All petitions are divided into four different classes based on the number of signatures N. (upper left) Distribution of the local variation for the real signing activity spike train data. (upper right) Same as the latter for randomized spike trains (null model), showing behavior that is more clearly Poissonian and the same for all classes. (lower left) The mean μ(L_V) of real and randomized spike trains for different classes of numbers of signatures. (lower right) The z-values of real and randomized data for different classes of numbers of signatures, showing that the classes with only a few signatures deviate from the Poissonian assumption according to the L_V measure.

Fig 5. Local variation analysis of petition signing spike trains for different duration classes.

All petitions are divided into eight different classes based on their duration T. A small class index corresponds to short durations and large one to long durations. (upper left) Distribution of the local variation for the real petition spike train data. (upper right) Same as the latter for randomized spike trains, showing behavior that is more clearly Poissonian and the same for all classes. (lower left) The mean μ(L_V) of real and randomized spike trains for different duration classes. (lower right) The z-values of real and randomized data for different petition duration classes.

Fig 6. Correlation of local variation for successive time intervals and the related Pearson correlation coefficient.

(left) Correlation plot for the local variation L_V for two successive time intervals. (right) The corresponding Pearson correlation coefficient for different classes of numbers of signatures.

Fig 7. Inter-event time distribution and burstiness.

(left) The PDF of the burstiness coefficient B and the corresponding mean values (inset). (right) The PDF of the memory coefficient M and the corresponding mean values (inset).

Fig 8. Memory coefficient and burstiness.

(left) Burstiness vs. memory coefficient. One clearly sees deviations from the human activity patterns in Ref. [10], where M is close to zero. (right) The local variation L_V vs. memory coefficient.