Cluster-size decay in supercritical kernel-based spatial random graphs

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(k<|C(0)|<∞)=exp(−Θ(kζ))ask→∞. 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((logn)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.