Ordering and Convergence of Large Degrees in Random Hyperbolic Graphs

Abstract: We describe the asymptotic behaviour of large degrees in random hyperbolic graphs, for all values of the curvature parameter $ \alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the scale-free regime $ \alpha>1/2$, the rank at which these two rankings cease to coincide is $n^{1/(1+8 \alpha)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $ \alpha = 1/2$, which corresponds to the connectivity threshold of the model. For $ \alpha < 1/2$, the maximum degree is of order $n - O(n^{ \alpha + 1/2})$, whereas for $ \alpha \geq 1/2$, the maximum degree is of order $n^{1/(2 \alpha)}$. In the cases $ \alpha < 1/2$ and $ \alpha > 1/2$, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $ \alpha < 1/2$ and Fr\'echet for $ \alpha > 1/2$). This refines previous estimates on the maximum degree for $ \alpha > 1/2$ and extends the study of large degrees to the dense regime $ \alpha \leq 1/2$.