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Various different random graph models have been proposed in which thevertices of the graph are seen as members of a metric space, and edges betweenvertices are determined as a function of the distance between the correspondingmetric 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 considerthe case that $X$ is the $n\times n \times \dots\times n$ integer lattice indimension $r$, 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 andconnectivity in this model. We compare this model with a traditional Waxmangraph. Further, we discuss expected degrees of nodes in detail for dimension 2.

Connectivity and giant component in random distance graphs