Curvature and temperature of complex networks

We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the internet into the hyperbolic plane and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of internet routing.

Figure 1

(Color online) Left: artistic visualization of the Poincaré disk model of the hyperbolic plane ${\mathbb{H}}^2$ by Levy based on Escher’s Circle Limit III with the permission from the Geometry Center, University of Minnesota. The exponential expansion of fish illustrates the exponential expansion of hyperbolic space. All fish are of the same hyperbolic size but their Euclidean size exponentially decreases, while their number exponentially increases with the distance from the origin. Right: a modeled network with $N=740$ nodes, power-law exponent $\gamma =2.2$, and average degree $\overline{k}\approx 5$ embedded in the hyperbolic disk of curvature $K=-1$ and radius $R\approx 15.5$. The Euclidean distance between a node and the origin at the disk center, shown as the cross, represents the true hyperbolic distance between the two. But the Euclidean distance between any two other nodes is not equal to the hyperbolic distance between them as indicated by the peculiar shape of the shaded hyperbolic disk centered at the circled node located at distance $r=10.6$ from the origin. The hyperbolic radius of this disk is also $R$, and according to the model, the circled node is connected to all the nodes lying in this disk. The curves show the hyperbolically straight lines, i.e., geodesics, connecting the circled node and some nodes in its disk.

Figure 2

(Color online) Networks in the ${\mathbb{H}}^2$ model vs the internet. Left: the degree distribution $P\left(k\right)$ and degree-dependent clustering coefficient $\overline{c}\left(k\right)$ are shown for the skitter ($\overline{k}=6.29$, $\overline{C}=0.46$) and border gateway protocol (BGP) ($\overline{k}=4.68$, $\overline{C}=0.29$) views of the internet from [6] and for modeled networks with curvature $K=-0.83$ and two values of temperature $T$, 0.47 ($\overline{k}=6.03$, $\overline{C}=0.44$) and 0.71 ($\overline{k}=4.85$, $\overline{C}=0.25$). Right: the empirical connection probability in the hyperbolically embedded internet, compared to Eq. (5).