Abstract

In  this  article for  a finite typed  random geometric graph we define the empirical locality distribution, which records the number of nodes of a given type linked to a given number of nodes of each type.  We find large
deviation principle (LDP) for the \emph{ empirical locality measure}
given the empirical pair measure and  the empirical type measure of
the typed random geometric graphs. From this LDP, we derive large
deviation principles  for the \emph{degree measure and the proportion of detached nodes} in the classical Erd\H{o}s-R\'{e}nyi graph defined on $[0, 1]^d.$ This graphs have been suggested by (Canning and Penman, 2003) as a possible extension to the randomly typed random graphs.

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