Logarithmic scaling and stochastic criticality in collective attention

2026-01-18

AI Summary

This study uncovers universal scaling laws and an underlying stochastic critical mechanism governing collective attention dynamics. By analyzing large-scale Wikipedia pageview data, the authors formulate a stochastic differential equation model driven by fractional Brownian motion and demonstrate that the logarithmic variance of collective attention grows slowly with the logarithm of time, deviating from conventional power-law behavior. They introduce a critical boundary defined by a single exponent ξ = H − η, which separates power-law growth from saturation at ξ = 0, thereby classifying collective attention as a non-Markovian process characterized by long-range memory and ultraslow dynamics. Integrating a Gaussian mixture model, the framework successfully reproduces the observed logarithmic variance scaling and accurately reconstructs the cumulative attention distribution, revealing a profound connection to aging dynamics in glassy systems.

Technology Category

• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction
• Application Domains: Humanities & Computational Social Science
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• collective attention
• scaling law
• stochastic criticality
• logarithmic scaling
• non-Markovian processes

Innovation

• logarithmic scaling
• stochastic criticality
• fractional Brownian motion
• non-Markovian processes
• collective attention