Connectivity and giant component in random distance graphs

Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric space elements. We here propose a model $G=G(X, f)$, in which $(X, d)$ is a metric space, $V(G)=X$, and $\mathbb{P}(u\sim v) = f(d(u, v))$, where $f$ is a decreasing function on the set of possible distances in $X$. We consider the case that $X$ is the $n\times n \times \dots\times n$ integer lattice in dimension $r$, with $d$ the $\ell_1$ metric, and $f(d) = \frac{1}{n^\beta d}$, and determine a threshold for the emergence of the giant component and connectivity in this model. We compare this model with a traditional Waxman graph. Further, we discuss expected degrees of nodes in detail for dimension 2.