Year: 2025
Type: article
Abstract: We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter alpha α $\alpha$ . 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 sparse regime alpha greater than one half α > 1 2 $\alpha>\tfrac{1}{2}$ , the rank at which these two rankings cease to coincide is n Superscript 1 divided by left parenthesis 1 plus 8 alpha right parenthesis plus o left parenthesis 1 right parenthesis n 1 / ( 1 + 8 α ) + o ( 1 ) $n^{1/(1+8\alpha)+o(1)}$ . 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 equals one half α = 1 2 $\alpha = \tfrac{1}{2}$ , which corresponds to the connectivity threshold of the model. For alpha less than one half α < 1 2 $\alpha < \tfrac{1}{2}$ , the maximum degree is of order n minus upper O left parenthesis n Superscript alpha plus 1 divided by 2 Baseline right parenthesis n − O ( n α + 1 / 2 ) $n - O(n^{\alpha + 1/2})$
Source: Advances in Applied Probability
Author: Loïc Gassmann
Institution: University of Fribourg
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
Award: Dynamic hyperbolic graphs
OpenAlex ID: W4415565967