Iyer, Srikanth K (2018) The random connection model: Connectivity, edge lengths, and degree distributions. In: RANDOM STRUCTURES & ALGORITHMS, 52 (2). pp. 283300.

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Abstract
Consider the random graph G(Pn,r) whose vertex set Pn is a Poisson point process of intensity n on (12,12]d,d2. Any two vertices Xi,XjPn are connected by an edge with probability g(d(Xi,Xj)r), independently of all other edges, and independent of the other points of Pn. d is the toroidal metric, r > 0 and g:0,)0,1] is nonincreasing and =dg(x)dx <. Under suitable conditions on g, almost surely, the critical parameter Mn for which G(Pn,<bold></bold>) does not have any isolated nodes satisfies lim?nnMndlog?n=1. Let =inf?{x > 0:xg(x)> 1}, and be the volume of the unit ball in d. Then for all >, G(Pn,(log?nn)1d) is connected with probability approaching one as n. The bound can be seen to be tight for the usual random geometric graph obtained by setting g=10,1]. We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy g(r)=o(rc).
Item Type:  Journal Article 

Publication:  RANDOM STRUCTURES & ALGORITHMS 
Additional Information:  Copy right for this article belong to the WILEY, 111 RIVER ST, HOBOKEN 070305774, NJ USA 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  02 Mar 2018 14:52 
Last Modified:  02 Mar 2018 14:52 
URI:  http://eprints.iisc.ac.in/id/eprint/59072 
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