# Stretched Exponential Dynamics in Online Article Views

^{1}Soft Matter Physics Laboratory, School of Advanced Materials Science and Engineering, SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon, South Korea^{2}Research Center for Advanced Materials Technology, Sungkyunkwan University, Suwon, South Korea

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.

Nowadays, many scientific journals promote online publication, which is efficient in rapidly and widely distributing scientific and public impact of scientific articles through online community. Once an open-access article is published on a journal website, the public can access the article immediately [1]. Article view statistics delivers useful statistical information of online published articles. Statistics of article page views is a useful alternative measure to quantify the impact of individual articles, independent of the journal impact factor [2–5]. Empirically, when an article receives attention, its daily page views would reach an initial peak and in turn diminish with time. We suppose that such a decay pattern in article page views would be similar to a relaxation process. On this basis, we assume there may be a popularity decay pattern for article page views, as suggested in a recent study on observations from six different PLoS journals [1]. It is a significant task to find out what mathematical model is appropriate for statistics in article page views.

In physics, a relaxation dynamics is a spontaneous evolution of out-of-equilibrium systems [6]. Glasses, for instance, usually show a relaxation dynamics toward equilibrium, which is commonly referred to as a physical aging [7]. Nonexponential relaxation dynamics, deviated from a simple exponential decay, ubiquitously appears as a relaxation dynamics in complex systems. Nonexponentiality in a relaxation dynamics is frequently described by a stretched exponential model [8–10] or the Kohlrausch-Williams-Watts (KWW) function [11, 12] (also known as the Weibull function [13]), expressed as *s*) monotonically decreases with time (*t*). This function is determined by two specific parameters: α is the characteristic lifetime corresponding to the specific time for

In this article, we utilize reliable statistical data for article page views that are available for online published articles in *Scientific Reports* and particularly we collect articles that were published after January 1, 2012. Data for daily page views including HTML views and PDF downloads were offered 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

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 *p* [*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

First, we counted the number of page views (*t* (days) that can be normalized by the number of initially maximized page views (*s* vs. *t* is invalid in popularity dynamics of online articles.

**FIGURE 1**. Statistics of article page views after online publication. The normalized page views are described in the normal scales **(A,B)** and in the lognormal scales **(C,D)**. The first dataset is taken from 42 articles published in *Scientific Reports* in 2012 **(A,C)** and the second dataset from 19 articles published in *Scientific Reports* between January and May in 2015 **(B,D)**, showing more than 1,000 views at the early stage of online publication. As a guide, the arbitrary selected data for srep00223 **(A,C)** and srep07971 **(B,D)** are marked with the solid line, respectively. Most interestingly, fast decays in the early stage and slow decays in the late stage can be observed.

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 *p* [*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

**FIGURE 2**. Time dependence of stretched exponents. The time-dependent stretched exponents **(A,B)**. Based on the fitting of **(C,D)**, while the biexponential function (the black line) is inappropriate to the long-tailed popularity dynamics. For a guide, srep00223 **(A,C)** and srep07971 **(B,D)** are marked with the solid lines.

Based on the estimation for *s* curve (the bold line) agrees with the real data (the solid line) curve quite well, respectively. Particularly, the long-tailed popularity dynamics (

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 [22–25]. 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* curve (the bold line) (because of good fitting with high adj.

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 not-so-popular articles because their peaks are weak. Further studies are required to search for a universal decay pattern in page views.

Long-time (

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.

## Funding

This manuscript has been released as a pre-print at (https://arxiv.org/abs/1606.04972) (34). This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B01007133, 2019R1A6A1A03033215) and also supported by the Korea Evaluation Institute of Industrial Technology funded by the Ministry of Trade, Industry, and Energy (20000423, Developing core technology of materials and processes for control of rheological properties of NanoInk for printed electronics) and by the MOTIE (Ministry of Trade, Industry, and Energy) in Korea, under the Fostering Global Talents for Innovative Growth Program (P0008746) supervised by the Korea Institute for Advancement of Technology (KIAT).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2020.619729/full#supplementary-material

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Keywords: stretched exponential, page view, online publication, popularity, relaxation

Citation: Kim Y and Weon BM (2021) Stretched Exponential Dynamics in Online Article Views. *Front. Phys.* **8**:619729. doi: 10.3389/fphy.2020.619729

Received: 21 October 2020; Accepted: 27 November 2020;

Published: 14 January 2021.

Edited by:

Wei-Xing Zhou, East China University of Science and Technology, ChinaReviewed by:

Bikas K. Chakrabarti, Saha Institute of Nuclear Physics (SINP), IndiaHaroldo V. Ribeiro, State University of Maringá, Brazil

Copyright © 2021 Kim and Weon. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Byung Mook Weon, bmweon@hotmail.com