The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs

We study a new kind of proximity graphs called proportional-edge proximity catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random catch digraphs that have been developed recently and have applications in statistical pattern classification and spatial point pattern analysis. PCDs are also a special type of intersection digraphs; and for one-dimensional data, the proportional-edge PCD family is also a family of random interval catch digraphs. We present the exact (and asymptotic) distribution of the domination number of this PCD family for uniform (and non-uniform) data in one dimension. We also provide several extensions of this random catch digraph by relaxing the expansion and centrality parameters, thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe sudden jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter values.Comment: 29 pages, 3 figure

Extension of One-Dimensional Proximity Regions to Higher Dimensions

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples

Distribution of the Relative Density of Central Similarity Proximity Catch Digraphs Based on One Dimensional Uniform Data

We consider the distribution of a graph invariant of central similarity proximity catch digraphs (PCDs) based on one dimensional data. The central similarity PCDs are also a special type of parameterized random digraph family defined with two parameters, a centrality parameter and an expansion parameter, and for one dimensional data, central similarity PCDs can also be viewed as a type of interval catch digraphs. The graph invariant we consider is the relative density of central similarity PCDs. We prove that relative density of central similarity PCDs is a U-statistic and obtain the asymptotic normality under mild regularity conditions using the central limit theory of U-statistics. For one dimensional uniform data, we provide the asymptotic distribution of the relative density of the central similarity PCDs for the entire ranges of centrality and expansion parameters. Consequently, we determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We also provide the connection with class cover catch digraphs and the extension of central similarity PCDs to higher dimensions.Comment: 28 pages, 6 figure