基于随机有向图族的图的相对边密度分布
Distribution of Relative Edge Density of the Graphs Based on a Random Digraph Family
摘要：
有向邻近捕捉图是近年来提出的一种顶点随机图，在模式识别和空间模式分析中有着广泛的应用。PCD是随机有向图（即，有向图），该有向图是使用来自各种类的点的相对位置从数据构造的。不同的PCD由与每个数据点相关联的邻近区域的不同定义产生。我们考虑的基础和自反图的基础上的一个家庭的PCD，这是由一个家庭的参数化的接近地图称为比例边（PE）接近地图。我们研究的图不变量是底层图和自反图的相对边密度。我们证明，适当缩放，这些图的相对边缘密度是一个$U$ -统计量，因此获得的渐近正态性的相对边缘密度的数据从任何分布，满足温和的监管条件。通过详细的概率计算和几何计算，我们计算了通常欧氏平面上有界区域上均匀数据的渐近正态分布的显式形式。我们还比较了两种图的相对边密度和PE-PCD的相对弧密度。这里提出的方法也适用于更高维度的数据。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
