Connectivity of networks with general connection functions

In the original Gilbert model of random geometric graphs, nodes are placed according to a Poisson process, and links formed between those within a xed range. Motivated by wireless network applications \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 directly connect. In many applications, not only in wireless networks, it is desirable that the graph is fully connected, that is every node is connected to every other node in a multihop fashion. Here, the full connection probability of a dense network in a convex polygonal or polyhedral domain 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 connection functions used in previous studies and with numerical simulations.

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