Stretched Exponential Dynamics in Online Article Views

Article view statistics offers a measure to quantify scientific and public impact of online published articles. Popularity of a paper in online community changes with time. To understand popularity dynamics of article views, we propose a decay dynamics based on a stretched exponential model. We find that a stretched exponent gradually decreases with time after online publication following a power-law scaling. Compared with a simple exponential or biexponential model, a stretched exponential model with a time-dependent exponent well describes long-tailed popularity dynamics of online articles. This result gives a useful insight into how popularity diminishes with time in online community.

48 h after online publication and updated daily. Here, we demonstrate a useful analysis methodology for evaluating nonexponentiality in article page views by applying the modified stretched exponential function, where the stretched exponent is modified to be time dependent and defined as β(t) in s exp[−(t/α) β(t) ] [14][15][16][17][18][19]. We find that the decay dynamics of article page views follows the modified stretched exponential model and particularly the stretched exponent gradually decreases with time following a power-law scaling. This finding suggests a possibility that popularity dynamics of online articles resembles a physical relaxation dynamics as can be frequently found in complex systems.
We collected the following two datasets: the first dataset comprised 42 articles from Scientific Reports published in 2012 and the second dataset comprised 19 articles published between January and May in 2015, showing more than 1,000 daily page views in a few days after online publication (see the supplementary tables). From the normalized page views described as s exp[−(t/α) β(t) ], the characteristic lifetime α can be measured by detecting the interception point between s(t) and s(α) exp(−1). The estimate of the α value can be obtained from a linear regression from two neighboring data points that exist just above (p + 1) and just below (p − 1) the p [ ≈ exp(−1)] point for each s vs. t curve [18,19]. The time-invariant (for the simple or the stretched exponential model) or the time-dependent stretched exponent (for the modified stretched exponential model) can be evaluated by identifying the time dependence of β(t) ln[−ln(s)]/ln(t/α).
First, we counted the number of page views (p t ) at time t (days) that can be normalized by the number of initially maximized page views (p 0 ) at t 0. The normalized popularity is defined from the normalized page views as s p t /p 0 . The initial popularity commonly decreases with time from the initial peak. For demonstration, the normalized page views for our two datasets are illustrated by the normal scales in Figures 1A Figures 1A,B. Here, the straightness in the logarithmic scales is likely to change with time, suggesting a combination of a fast initial decay and a slow late decay. The possibility of two different time scalings is consistent with the previous observation [1]. The combination of fast and slow dynamics is manifested in the biexponential decay that is determined by two different time constants [20]. This result implies that a single power-law scaling of s vs. t is invalid in popularity dynamics of online articles.
Next, we tested the nonexponentiality in the normalized page views. To examine the nonexponentiality, the most feasible methodology is the application of the modified stretched exponential function s exp[−(t/α) β(t) ]. This function is quantified with the characteristic lifetime α and the timedependent stretched exponent β(t) [14,15]. The modified stretched exponential function was invented to describe abnormal photoluminescence dynamics and human survival curves [16][17][18][19]. In general, the time-invariant (for the simple or the stretched exponential model) or the time-dependent stretched exponent (for the modified stretched exponential model) can be evaluated by identifying the time dependence of β(t) ln[−ln(s)]/ln(t/α). The characteristic lifetime α can be measured by detecting the interception point between s(t) and s(α) exp(−1). The estimate of the α value can be obtained from a linear regression from two neighboring data points that exist just above (p + 1) and just below (p − 1) the p [ ≈ exp(−1)] point for each s vs. t curve [18,19]. Here the time dependence of the stretched exponent is a good measure for testing the nonexponentiality of a decay curve and relevant to the relaxation time distribution [9]. As illustrated in Figures 2A,B, we are able to evaluate the nonexponentiality of the normalized page views for two different datasets published in Scientific Reports. Definitely, the stretched exponent significantly deviates from unity as β(t) ≠ 1, indicating the nonexponentiality in the normalized popularity of online articles; that is, the simple exponential model is not valid. Most importantly, the stretched exponent decreases with time, showing a power-law time dependence for the lifetime after initial irregular changes. The long-tailed power-law time dependence of the β(t) value can be formulated as β(t) ct δ (t > 10). From the representative data (srep00223 and srep07971) t > 10, c 0.8096 ± 0.0132 and δ −0.1564 ± 0.0027 (adj. R 2 0.7049) in Figure 2A and c 0.9589 ± 0.029 and δ −0.1807 ± 0.0064 (adj. R 2 0.7355) in Figure 2B were estimated by a powerlaw fitting, as described by the bold line, respectively.
Based on the estimation for β(t) ct δ in Figures 2A,B (the bold line), we are able to calculate the normalized page views with s exp[−(t/α) β(t) ], which reveals the popularity dynamics of online articles. For two representative datasets, as demonstrated in Figures 2C,D, the calculated s curve (the bold line) agrees with the real data (the solid line) curve quite well, respectively. Particularly, the long-tailed popularity dynamics (t > 100) is well described by this model.
We discuss a decay model in popularity dynamics of online articles. In fact, characterizing and modeling popularity dynamics is a difficult task, because online community consists of complex networks [21]. At any rate, popularity dynamics is likely to have a decay dynamics that consists of a short-term rapid decay and a long-tailed slow decay as expected from data in the literature [22][23][24][25]. On this basis, a recent study on article citation dynamics suggests a biexponential function, which fits the data better than the log-normal and the simple exponential models [20]. In the biexponential model, s A 1 exp[−(t/τ 1 )] + A 2 exp[−(t/τ 2 )], there must exist two different time constants τ 1 and τ 2 . The coefficients A 1 and A 2 represent the relative contribution of two exponential decays. The biexponential function well explains the popularity dynamics including article citation, music, movie, and website page view for t < 100 [20]. In Figures 2C,D, we incorporate the biexponential function to the calculated s curve (the bold line) (because of good fitting with high adj. R 2 > 0.96) instead of the real data (because of bad fitting with low adj. R 2 < 0.86). Interestingly, for short times (t < 50), the biexponential function (the black line) is equal to our model (the bold line) but significantly deviates from the data and the modeled curve for long times (t > 50). This analysis indicates that the biexponential decay is appropriate for the short-term dynamics but inappropriate for the long-tailed dynamics in popularity dynamics. Now, we presume that the origin of the modified stretched exponential function is a combination of multiple exponential functions that consist of multiple time constants as s Our finding suggests that the relaxation dynamics in article page views resembles the physical relaxation dynamics as can be frequently found in complex systems. Social media are indeed complex systems that actively connect people every day. Interestingly, the temporal evolution of the stretched exponents in the normalized page views is identical in the luminescence decays, where the stretched exponents that are smaller than unity gradually decrease with time (see Refs. 14 and 15). The relaxation dynamics of public attention and physical excitation at the initial period would be alike with respect to the time evolution of impact and information propagation. A stochastic model for information propagation would be relevant to the nonexponentiality of page view statistics [1]. Our finding from selected datasets needs to be generalized from diverse datasets. Our analysis may not be suitable to describe the page view dynamics of notso-popular articles because their peaks are weak. Further studies are required to search for a universal decay pattern in page views.
Long-time ( ∼ years) citation statistics are often alike with short-time ( ∼ days) page view statistics: there would be unpopular, popular, or revived articles (sleeping beauties) [26][27][28]. Scientific papers typically have a finite lifetime: their rate to attract citations achieves its maximum a few years after publication and then steadily declines [28]. The overall citation decays can be power-law or exponential [29][30][31]. We believe that the citation dynamics and the aging effect in citation networks [31][32][33] would be evaluated with the modified stretched exponential model, similar to the page view dynamics.
In conclusion, we present a useful statistical approach for article page views of online published articles. A significant finding is obtained: article page views usually decay over time after reaching initial peaks, especially exhibiting the nonexponentiality. A feasible methodology based on the stretched exponentiality is suggested for evaluation of the nonexponentiality. To understand popularity dynamics of article views, we propose a popularity decay dynamics based on a stretched exponential model. We find that a stretched exponent gradually decreases with time after online publication following a power-law scaling. Compared with a biexponential model, a stretched exponential model with a time-dependent exponent well describes long-tailed popularity dynamics of online articles. This result shows a general insight into how popularity decreases with time in online community.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
YK and BW organized and conducted the research, analyzed the data, and wrote the manuscript.