On the evolution of random graphs on spaces of negative curvature

In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points, with the weight being proportional to a real parameter \beta>0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. This class of random graphs was introduced by Krioukov et al. We provide a rigorous analysis of aspects of this model and its dependence on the parameter \beta, verifying some of their observations. We show that a phase transition occurs around \beta =1. More specifically, we show that when \beta > 1 the degree of a typical vertex is bounded in probability (in fact it follows a distribution which for large values exhibits a power-law tail whose exponent depends only on the curvature of the space), whereas for \beta <1 the degree is a random variable whose expected value grows polynomially in N. When \beta = 1, we establish logarithmic growth. For the case \beta > 1, we establish a connection with a class of inhomogeneous random graphs known as the Chung-Lu model. Assume that we use the Poincar'e disc representation of a hyperbolic space. If we condition on the distance of each one of the points from the origin, then the probability that two given points are adjacent is expressed through the kernel of this inhomogeneous random graph.

💡 Research Summary

The paper provides a rigorous mathematical treatment of a family of random geometric graphs defined on hyperbolic spaces, a model originally introduced by Krioukov et al. N points are sampled in a hyperbolic disc (using the Poincaré disc representation) according to a radial density that reflects the negative curvature. For any pair of points i and j, an edge is placed independently with probability p_{ij}=g(β,d_{ij}), where d_{ij} is the hyperbolic distance and β>0 plays the role of an inverse temperature. This probability can be written in a Boltzmann form, so that, conditioned on the point locations, the graph distribution is a Boltzmann distribution with Hamiltonian equal to β times the sum of distances over all present edges.

The authors focus on how the parameter β controls the global structure of the resulting network. They prove three distinct regimes separated by a critical value β_c=1.

1. Low‑temperature regime (β>1). The degree of a typical vertex remains bounded in probability; its distribution exhibits a power‑law tail P(D>k)∼k^{-τ} where the exponent τ depends only on the curvature of the space and on β. Moreover, when the radial distances of the vertices are fixed, the edge probability can be expressed as a kernel of an inhomogeneous random graph of the Chung‑Lu type: each vertex i receives a weight w_i≈e^{(R−r_i)/2}, and p_{ij}=w_i w_j/∑_k w_k up to negligible errors. Consequently, all known results for Chung‑Lu graphs (e.g., component sizes, clustering, typical distances) transfer to this hyperbolic model.

2. High‑temperature regime (β<1). The expected degree grows polynomially with the number of vertices: E