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 behavior in the high-citation regime-an apparent contradiction lacking a unified explanation. Here we identify a previously unrecognized 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, modeled 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) favor 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≈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.