Graph connectivity with fixed endpoints in the random-connection model
arxiv(2023)
摘要
We consider the count of subgraphs with an arbitrary configuration of
endpoints in the random-connection model based on a Poisson point process on
${\Bbb R}^d$. We present combinatorial expressions for the computation of the
cumulants and moments of all orders of such subgraph counts, which allow us to
estimate the growth of cumulants as the intensity of the underlying Poisson
point process goes to infinity. As a consequence, we obtain a central limit
theorem with explicit convergence rates under the Kolmogorov distance, and
connectivity bounds. Numerical examples are presented using a computer code in
SageMath for the closed-form computation of cumulants of any order, for any
type of connected subgraph and for any configuration of endpoints in any
dimension $d\geq 1$. In particular, graph connectivity estimates, Gram-Charlier
expansions for density estimation, and correlation estimates for joint subgraph
counting are obtained.
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