Logarithmic scaling and stochastic criticality in collective attention

TL;DR

This paper reveals a universal logarithmic scaling law in collective attention dynamics, showing that attention dispersion grows ultraslowly due to stochastic criticality, modeled by fractional Brownian motion with long-range memory.

Contribution

It introduces a minimal stochastic differential equation model capturing the ultraslow growth of attention variance and identifies the critical boundary between different dynamical regimes.

Findings

Variance of attention grows logarithmically over time.

The model links long-range memory and volatility decay through a single exponent.

Empirical data from Wikipedia supports the stochastic criticality framework.

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 ($\eta$) enter through the single exponent $\xi\equiv H-\eta$. At marginality, $\xi=0$, the variance grows logarithmically, marking the critical boundary between power-law growth ($\xi>0$) and saturation ($\xi<0$). 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.