Fig 1.
Circadian pattern of tweeting activity.
Increasing amount of tweets from midday (12:00) to midnight (00:00) is shown in the yellow shaded regions. Significant decays of the activity are observed during nights. The activity increases during mornings as shown in purple shaded rectangles. In the inset, we show the temporal evolution at a finer scale, where fluctuations are visible. The data exhibit two peaks: The first one is in the evening of a political debate, on May 2 2012 and the second is on the French presidential election day, May 6 2012.
Table 1.
The first 40 most used hashtags are listed with the corresponding popularity p. The hashtags related to the debate and the presidential election such as ledebat, hollande, sarkozy, votehollande, france2012, and présidentielle are recognized.
Fig 2.
Heterogeneity in the hashtag popularity p is shown in (a) Zipf-plot and (b) probability density function (PDF), P(p).
(a) Diversity in p (frequency) is visible in a power-law scaling in the log-log plot. We rank hashtags from high p (left) to low p (right). Different colored shaded rectangles highlight the value of p from red and orange (high p) to purple and pink (low p). The percentages describe the overall contributions of the corresponding rectangles. (b) Similarly, P(p) obeys a slowly decaying function and presents a power-law distribution with a fat tail. The same colored schema in (a) is applied to visualize the contributions of different values of p.
Fig 3.
The cumulative (a), CDF(Δτ), and probability (b), P(Δτ), distributions of the inter-hashtag spike intervals.
We observe that P(Δτ), for different classes of hashtags distinguished by their popularity, exhibits non-exponential features. The different colors correspond to those in Fig 2. The legend provides the average popularity ⟨p⟩ in each hashtag class. The dash lines indicate the positions of 1 day, 2 days, and 3 days, where P(Δτ) gives peaks for low p (pink symbols). The binning is varied from 8 minutes to 2 hours depending on p, e.g. 8 min. for high p (red-orange), 1.5 hour for moderate p (yellow-green-blue-purple), and 2 hours for low p (pink). All P(Δτ) present maxima at 1 second, which is not shown to describe tails in a larger window.
Fig 4.
Real and artificial hashtag spike trains.
(a) As an illustration of different hashtag spike trains representing different types of hashtag propagation of the data set. (b) Merging hashtag spike trains from the real data. The black spikes describe that only one activity is counted if multiple activities occur at the same time. (c) Randomization procedure by randperm (Matlab). T contains full hashtag activity of the data set. The randperm gives a matrix with p elements, p unique independent numbers out of T, and constructing random time series …, ,
,
, … from full hashtag activity matrix T. (d) The resultant artificial hashtag spike train.
Fig 5.
The probability distribution of count of hashtag activity per second P(ch).
We show that, except for the top most popular hashtags listed in Table 1 with ranking 1–11 and presented here in red symbols, multiple activity in 1 second is very rare. The different colors correspond to those in Figs 2 and 3. The legend provides the average popularity ⟨p⟩ in each hashtag class.
Fig 6.
The local variation LV of hashtag spike trains versus popularity p on a linear-log plot.
Each color and symbol summarized in the legend present different range of p: Low p, pink and purple colors, and moderate p, blue, green, and yellow colors, and then high p, orange and red colors. In addition, the average p, ⟨p⟩, indicated in the legend ranks colors and symbols quantitatively. (a) Hashtag spike trains of the data set. (b) Artificial (randomized) hashtag spike trains.
Fig 7.
Probability density function (PDF) of the local variation LV of real hashtag propagation (a) and random hashtag time sequences (b).
Two distinct shapes are visible: (a) From high p to low p, the peak position of P(LV) shifts from low values of LV to higher values of LV. (b) P(LV) always peaks around 1 for the random sequences generated by artificial hashtag spike trains. The same color coding is applied as already used in Fig 6.
Fig 8.
Statistical inference of LV and comparison between the real and the random hashtag spike trains.
(a) Mean μ of the local variation LV of single hashtag time series versus the logarithmic average popularity log10⟨p⟩. The real hashtag propagation is described in blue circles, whereas red squares represent randomly selected hashtag activity from the real data set. The arrow indicates the decay of μ(LV) when ⟨p⟩ increases, which shows that popular hashtags propagate regularly on the contrary to moderately popular hashtags presenting bursty time sequences. The bars indicate the corresponding standard deviations σ(LV). (b) A standard z−values versus log10⟨p⟩. While the random trains (red squares) with z ≈ 0 show the evidence of Poisson signals with mean μ0(LV) = 1, large and non-zero values of z for the real trains (blue circles) suggest the presence of temporal correlations.
Fig 9.
Linear correlation of LV through real hashtag spike trains.
(a) The linear relations of the first and the second halves of the empirical spike trains, LV(t1) and LV(t2), respectively, are investigated. The legend ranks ⟨p⟩ in different colors and symbols. (b) The Pearson correlation coefficient r(LV(t1), LV(t2)) between these quantities shows that while the linear correlations through moderately popular spike trains give maximum values, r reaches the minimum values for both bursty (high LV and low p) and regular (low LV and high p) spike trains.