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Abstract
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.
Original language  English 

Article number  032313 
Number of pages  14 
Journal  Physical Review E 
Volume  93 
Issue number  3 
Early online date  14 Mar 2016 
DOIs  
Publication status  Published  Mar 2016 
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Professor Carl P Dettmann
 Probability, Analysis and Dynamics
 School of Mathematics  Professor of Applied Mathematics
 Mathematical Physics
Person: Academic , Member, Group lead