Poisson approximation and connectivity in a scale-free random connection model

Abstract

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process Ps of intensity s > 0 on the unit cubeEach vertex is endowed with an independent random weight distributed as W , whereGiven the vertex set and the weights an edge exists between x, y ∈ Ps with probability 1 − exp − ηWxWy (d(x,y)/r) α , independent of everything else, where η, α > 0, d(•, •) is the toroidal metric on S and r > 0 is a scaling parameter. We derive conditions on α, β such that under the scaling rs(ξ) d = 1 c 0 s log s + (k − 1) log log s + ξ + log αβ k!d , ξ ∈ R, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e −ξ as s → ∞, where c0 is an explicitly specified constant that depends on α, β, d and η but not on k. In particular, for k = 0 we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large s. The Poisson approximation result is derived using the Stein's method.