  In the original (1961) Gilbert model of random geometric graphs, nodes are
placed according to a Poisson point process, and links formed between those
within a fixed range. Motivated by wireless ad-hoc networks "soft" or
"probabilistic" connection models have recently been introduced, involving a
"connection function" H(r) that gives the probability that two nodes at
distance r are linked (directly connect). In many applications (not only
wireless networks), it is desirable that the graph is connected, that is every
node is linked to every other node in a multihop fashion. Here, the connection
probability of a dense network in a convex domain in two or three dimensions is
expressed in terms of contributions from boundary components, for a very
general class of connection functions. It turns out that only a few quantities
such as moments of the connection function appear. Good agreement is found with
special cases from previous studies and with numerical simulations.

Statistical Mechanics;Networking and Internet Architecture