Distribution of Relative Edge Density of the Graphs Based on a Random Digraph Family

Abstract: The vertex-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed from data using the relative positions of the points from various classes. Different PCDs result from different definitions of the proximity region associated with each data point. We consider the underlying and reflexivity graphs based on a family of PCDs which is determined by a family of parameterized proximity maps called proportional-edge (PE) proximity map. The graph invariant we investigate is the relative edge density of the underlying and reflexivity graphs. We demonstrate that, properly scaled, relative edge density of these graphs is a $U$-statistic, and hence obtain the asymptotic normality of the relative edge density for data from any distribution that satisfies mild regulatory conditions. By detailed probabilistic and geometric calculations, we compute the explicit form of the asymptotic normal distribution for uniform data on a bounded region in the usual Euclidean plane. We also compare the relative edge densities of the two types of the graphs and the relative arc density of the PE-PCDs. The approach presented here is also valid for data in higher dimensions.