47 research outputs found
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