Fundamental dynamics of popularity-similarity trajectories in real networks

TL;DR

It is shown 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≪0.5, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks.

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.

Figures (25)

• Figure 1: Properties of the popularity trajectory of CLT in the US Air (in blue). (a) Radial popularity trajectory. (b) Trajectory increments (velocity). (c) Sample autocorrelation of the velocity. (d) Variance-time plot of the velocity. (e) Probability density function (pdf) of the velocity. The dashed curve shows a Gaussian pdf with the same mean and variance. (f) Expected degree trajectory. The estimated Hurst exponents for the trajectories in (a) and (f) are $0.0004$ and $0.02$. The plots show also results for a simulated counterpart of the trajectory (in red) constructed using the model of Eq. (\ref{['eq:rlfbm_main']}).
• Figure 2: Properties of the similarity trajectory of CLT in the US Air (in blue). (a)-(e) show the same as (a)-(e) in Fig. \ref{['pop_clt']}, but for the similarity trajectory. The $y$-axis in (a) is in radians. (f) Angular distribution of the trajectory. The estimated Hurst exponent for the trajectory is $0.03$.
• Figure 3: Distribution of Hurst exponents in US Air and IPv6 (in blue). (a) and (b) correspond to the expected degree trajectories. The average Hurst exponents are respectively $0.01$ and $0.13$. (c) and (d) correspond to the similarity trajectories. The average Hurst exponents are $0.08$ and $0.25$. We consider trajectories with at least $300$ points. The distributions for the randomized counterparts are shown in yellow. Similar results hold for the radial trajectories (Appendix \ref{['sec:radials']}).
• Figure 4: Expected degree and similarity trajectories in synthetic temporal networks constructed according to RHGs, the CM, and RGs. (a) and (b) show the distribution of Hurst exponents for the expected degree and similarity trajectories in the three cases. (c) and (d) show a similarity trajectory in RHGs and its distribution in the similarity space. The red line in (c) indicates the underlying attractor, i.e., the node's hidden similarity coordinate $\theta_h=4.9$. (e) and (f) show the same as in (c) and (d), but for the CM case. There is no underlying attractor here. The Hurst exponents for the trajectories in (c) and (e) are respectively 0.16 and 0.42.
• Figure 5: (a) Number of nodes, (b) average degree, and (c) average clustering in the US Air. We consider data up to time 10000 (May 20, 2015) indicated by the vertical line in the plots.