Fundamental dynamics of popularity-similarity trajectories in real networks

TL;DR

This paper reveals that node trajectories in hyperbolic embeddings of real networks exhibit universal self-similar, anti-persistent behaviors that can be modeled by fractional Brownian motion, linking dynamics to underlying geometry.

Contribution

It introduces a mathematical framework connecting the dynamics of real networks to their latent geometric space, demonstrating predictable, subdiffusive behavior.

Findings

Trajectories show universal self-similar properties with low Hurst exponents.

Behavior can be modeled by fractional Brownian motion.

Such dynamics are linked to the networks' hidden geometric structure.

Abstract

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics -- a grand-challenge open problem. Here, we show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents $H \ll 0.5$. This means that the trajectories are predictable, displaying anti-persistent or 'mean-reverting' behavior, and they can be adequately captured by a fractional Brownian motion process. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. These results set the foundations for rigorous mathematical machinery for describing and predicting real network dynamics.