  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.
