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.

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

References in corpus (7)

• Power-law distributions in empirical data
• Universality of citation distributions: towards an objective measure of scientific impact
• Defining and identifying Sleeping Beauties in science
• Nonuniversal power law scaling in the probability distribution of scientific citations
• Growing complex network of citations of scientific papers -- measurements and modeling
• Stochastic dynamical model of a growing network based on self-exciting point process
• Citation count distributions for large monodisciplinary journals