The random connection model: Connectivity, edge lengths, and degree distributions

Consider the random graph G(Pn,r) whose vertex set Pn is a Poisson point process of intensity n on (-12,12]d,d2. Any two vertices Xi,XjPn are connected by an edge with probability g(d(Xi,Xj)r), independently of all other edges, and independent of the other points of Pn. d is the toroidal metric, r > 0 and g:0,)0,1] is non-increasing and =dg(|x|)dx ) does not have any isolated nodes satisfies lim?nnMndlog?n=1. Let =inf?{x > 0:xg(x)> 1}, and be the volume of the unit ball in d. Then for all >, G(Pn,(log?nn)1d) is connected with probability approaching one as n. The bound can be seen to be tight for the usual random geometric graph obtained by setting g=10,1]. We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy g(r)=o(r-c).