Cluster-size decay in supercritical kernel-based spatial random graphs
Joost Jorritsma (University of Oxford)
Júlia Komjáthy (TU Delft - Applied Probability)
Dieter Mitsche (Université de Lyon, Pontificia Universidad Católica de Chile, University Jean Monnet)
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Abstract
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 (Formula presented) denoting the number of vertices in the component of the vertex at (Formula presented), we prove (Formula presented) 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 (Formula presented) 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.