Fractional stochastic model of citation dynamics with memory and volatility

Abstract

Understanding the statistical laws governing citation dynamics remains a fundamental challenge in network theory and the science of science. Citation networks typically exhibit in-degree distributions well approximated by log-normal distributions yet also display power-law behaviour in the high-citation regime -- an apparent contradiction lacking a unified explanation. Here we identify a previously unrecognised phenomenon: the variance of the logarithm of citation counts per unit time follows a power law with respect to time ($t$) since publication, scaling as $t^H$, with $H$ constant. This discovery introduces a new challenge while simultaneously offering a crucial clue to resolving this discrepancy. We develop a stochastic model in which latent attention to publications evolves through a memory-driven process with cumulative advantage, modelled as fractional Brownian motion with Hurst parameter $H$ and volatility. We show that antipersistent fluctuations in attention ($H<1/2$) yield log-normal citation distributions, whereas persistent attention dynamics ($H>1/2$) favour heavy-tailed power laws, thus resolving the log-normal--power-law contradiction. Numerical simulations confirm both the $t^H$ law and the transition between regimes. Empirical analysis of arXiv e-prints indicates that the latent attention process is intrinsically antipersistent ($H\approx 0.13$). By linking memory effects and stochastic fluctuations in attention to broader network dynamics, our findings provide a unifying framework for understanding the evolution of collective attention in science and other attention-driven processes.

arXiv commentMain Text: 14 pages (5 figures, 1 table); Supplementary Materials: 9 pages (6 figures, 2 tables)

References in corpus (32)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• Power-law distributions in empirical data
• Evolution of the social network of scientific collaborations
• How Popular is Your Paper? An Empirical Study of the Citation Distribution
• Self-similarity of complex networks
• Effect of Trends on Detrended Fluctuation Analysis
• Effect of nonstationarities on detrended fluctuation analysis
• Quantifying Long-Term Scientific Impact
• Universality of citation distributions: towards an objective measure of scientific impact
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Origins of fractality in the growth of complex networks
• Measuring preferential attachment for evolving networks
• Defining and identifying Sleeping Beauties in science
• Citation Statistics from 110 Years of Physical Review
• Highly clustered scale-free networks
• Characterizing and modeling citation dynamics
• The first-mover advantage in scientific publication
• Attention decay in science
• Citation Networks in High Energy Physics
• Nonuniversal power law scaling in the probability distribution of scientific citations
• Predicting the long-term citation impact of recent publications
• Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process
• Growing complex network of citations of scientific papers -- measurements and modeling
• Modelling Aging Characteristics in Citation Networks
• Stochastic dynamical model of a growing network based on self-exciting point process
• Universality of Performance Indicators based on Citation and Reference Counts
• The Time Dimension of Science: Connecting the Past to the Future
• Runaway Events Dominate the Heavy Tail of Citation Distributions
• Citation count distributions for large monodisciplinary journals
• Testing for universality of Mendeley readership distributions
• Scientometric engineering: Exploring citation dynamics via arXiv eprints