CS38: Random Geometric Systems

Organizer: Moritz Otto (Leiden University)

Ordering and convergence of large degrees in random hyperbolic graphs

Loïc Gassmann

A random hyperbolic graph is constructed by sampling \(n\) points in a hyperbolic disc and connecting pairs of points that are within a distance smaller than a certain threshold \(R_n\). When the curvature parameter \(\alpha\) is larger than \(1/2\) the graph is known to be a valid model for the so-called complex networks, which represent a broad class of real-world systems \[1\]. A complex networks is typically structured as follows: high-degree nodes are well connected to each other and serve as hubs for nodes with slightly lower degrees. These intermediate-degree nodes, in turn, connect to nodes with even lower degrees, and so on, forming a hierarchical structure.

The specificity of the random hyperbolic graph, compared to other complex networks models, is its embedding in a geometric space. The closer a node is to the centre of the underlying hyperbolic disc, the more important it tends to be in the network hierarchy. In this talk, we investigate this property precisely by comparing the ranking of the nodes by increasing distance to the centre with the ranking of the nodes by decreasing degree. We show that in the scale-free regime (\(\alpha > 1/2\)), the rank at which these two rankings cease to coincide is \(n^{1/(1+8\alpha)+o(1)}\), with high probability \[2\]. This rank marks the point beyond which spatial position no longer accurately reflects the importance of the nodes.

We also give a precise description of the largest degrees of the graph by stating the convergence in distribution of the point process of the degrees toward an explicit Poisson point process, for all positive values of the curvature parameter \(\alpha\). The transition at \(\alpha = 1/2\) is characterised by a maximum degree having a distribution out of the class of extreme value distributions \[2\].

Bibliography

\([1]\) Réka Albert and Albert-László Barabási, Statistical mechanics of complex networks, Rev. Modern Phys. 74 (2002), no. 1, 47-97

\([2]\) Loïc Gassmann, Ordering and convergence of large degrees in random hyperbolic graphs, 2025, preprint, arXiv:2404.06383

Large-Deviation Analysis for Canonical Gibbs Measures

Martina Petráková

Gibbs processes in the continuum are one of the most fundamental models in spatial stochastics. They are typically defined using a density with respect to the Poisson point process. In the language of statistical mechanics, this corresponds to the grand-canonical ensemble, where the number of particles is random. Of the same importance is the canonical ensemble, where the number of particles is fixed. In the language of point processes, this corresponds to studying binomial Gibbs processes which are defined using a density with respect to the binomial point process.

In this talk, we present a large deviation theory developed for functionals of binomial Gibbs processes with fixed intensity in increasing windows. Our method relies on the traditional large deviation result from \([1]\) noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions that allow us to handle delicate and unlikely pathological events. The presented results cover a broad class of both the interaction function (possibly unbounded) and the functionals (given as a sum of possibly unbounded local score functions).

Bibliography

\([1]\) Georgii, H.-O. and Zessin, H. (1993): Large deviations and the maximum entropy principle for marked point random fields, Probab. Theory Related Fields 96, 177–204.