Abstract
We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ2. From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]2 from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.
Highlights
cover catch digraph (CCCD) are generalized to random geometric digraphs called proximity catch digraphs (PCDs) in [4] and the minimum dominating sets of PCDs are proposed as solutions to the class cover problem (CCP) problem, and the cardinality of the minimum dominating sets of the PCDs is used as a test statistic for testing spatial clustering in a multi-class setting [4, 6]
We study the class cover problem (CCP) of random point sets in two dimensions
This goal is equivalent to finding a minimum dominating set for the digraph called the class cover catch digraph (CCCD)
Summary
The class cover problem (CCP) is motivated by its applications in pattern classification [18]. The balls around the members of the minimum dominating sets of the CCCDs can be used to construct discriminant regions for assigning class labels (see [8] for more detail) In this setting, we want to choose a class cover to represent class X that is as small as possible (i.e., a minimum dominating set for Xn), to make the classifier less complex while keeping most of the relevant information. CCCDs are generalized to random geometric digraphs called proximity catch digraphs (PCDs) in [4] and the minimum dominating sets of PCDs are proposed as solutions to the CCP problem, and the cardinality of the minimum dominating sets (i.e., domination number) of the PCDs is used as a test statistic for testing spatial clustering in a multi-class setting [4, 6]. The authors in [13, 27] study the percolation of the connectivity of these secrecy graphs under the lattice and Poisson models for the vertices
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