2023/03/01 by Joost Jorritsma, Júlia Komjáthy, Jorritsma, Joost +3 · 3 citations
Mathematics · Physics and Astronomy · #05C80 #60K35 #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2303.00724
openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent ζ∈(0,1) of the decay of the cluster-size distribution. That is, with |C(0)| denoting the number of vertices in the component of the vertex at 0∈ ℝd, we prove ℙ(klt; |C(0)|lt;∞)=exp(-Θ(kζ)), as k→∞. The value of ζ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent τ of the degree distribution and a long-range parameter α governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: ζ in this regime depends both on τ,α. We also prove that the second-largest component in a box of volume n is of size Θ((log n)1/ζ) with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space.