Logarithmic scaling and stochastic criticality in collective attention

TL;DR

The results place collective attention in a distinct class of non-Markovian stochastic processes, with close affinity to ageing-like and ultraslow dynamics in glassy systems.

Abstract

We uncover a universal scaling law governing the dispersion of collective attention and identify its underlying stochastic criticality. By analysing large-scale ensembles of Wikipedia page views, we find that the variance of logarithmic attention grows ultraslowly, $\operatorname{Var}[\ln{X(t)}]\propto\ln{t}$, in sharp contrast to the power-law scaling typically expected for diffusive processes. We show that this behaviour is captured by a minimal stochastic differential equation driven by fractional Brownian motion, in which long-range memory ($H$) and temporal decay of volatility ($η$) enter through the single exponent $ξ\equiv H-η$. At marginality, $ξ=0$, the variance grows logarithmically, marking the critical boundary between power-law growth ($ξ>0$) and saturation ($ξ<0$). By incorporating article-level heterogeneity through a Gaussian mixture model, we further reconstruct the empirical distribution of cumulative attention within the same framework. Our results place collective attention in a distinct class of non-Markovian stochastic processes, with close affinity to ageing-like and ultraslow dynamics in glassy systems.

Figures (7)

• Figure 1: Empirical evidence for marginal logarithmic scaling in collective attention (Wikipedia, 2020). (a) Median page views as a function of elapsed days $t$ since the reference day $d$, showing a slow decay with superposed weekly periodicity. (b) Increment-variance scaling following \ref{['eq:scaling_H']}, yielding an estimate of the Hurst exponent $H$. (c) Logarithmic growth of the variance $\operatorname{Var}[Y(t)]$ over $[0,t_\mathrm{c}]$, consistent with the marginal scaling law \ref{['eq:log_scaling']}. (d) Logarithmic decay of the mean $\mathbb{E}\!\left[Y(t)\right]$, implying a $1/t$ drift as in \ref{['eq:scaling_a']}. Data are based on Ref. Okamura25z, focusing on the top 1,000 viewed Wikipedia articles. The dotted vertical line in (c) and (d) marks $t=t_\mathrm{c}$, the upper bound of the fitting range.
• Figure 2: Histogram of the logarithm of cumulative page views over one year, overlaid with the fitted GMM. The data are based on Ref. Okamura25z, focusing on page views of the top 1,000 viewed articles on Wikipedia in 2020.
• Figure S1: (a) Autocorrelation function computed from the median page views as a function of elapsed days $t$ since the reference day (2020). Pronounced peaks appear at lag 7 days and its integer multiples, indicating the presence of a weekly periodic structure. (b) Periodogram in the frequency domain. A clear peak is observed at the frequency $1/7~\mathrm{day}^{-1}$, together with its harmonics, suggesting a non-sinusoidal weekly periodic pattern. Data are based on Ref. Okamura25z.
• Figure S2: Estimation of the Hurst exponent across years. Scatter plot of $\{\tau,\ln\operatorname{Var}[\Delta_{\tau}Y_{t}]\}$ for lags $\tau=2^{\ell}$ ($\ell=0,\dots,5$), together with a linear fit based on the scaling law in Eq. \ref{['eq:scaling_H']}. The slope yields an estimate of $2H$. Data are based on Ref. Okamura25z.
• Figure S3: Logarithmic growth of variance at marginality. Variance of $\ln(\text{page views}+1)$ as a function of elapsed days $t$. A linear fit over $[0,t_\mathrm{c}]$, based on the scaling law in Eq. \ref{['eq:scaling_gamma']}, demonstrates the ultraslow logarithmic growth $\operatorname{Var}[Y(t)]\propto\ln t$. The dotted vertical line marks $t=t_\mathrm{c}$. Data are based on Ref. Okamura25z.