We study the number of copies of a weakly connected subdigraph of the k nearest neighbor ( k NN) digraph based on data from certain random point processes in $$\mathbb {R}^d$$ R d . In particular, based on the asymptotic theory for functionals of point sets from homogeneous Poisson process (HPP) and uniform binomial process (UBP), we provide a general result for the asymptotic behavior of the number of weakly connected subdigraphs of k NN digraphs. As corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared k NN pairs, and the number of reflexive k NNs in the k NN digraph based on data from HPP and UBP. We also provide several extensions of our results pertaining to the k NN digraphs; more specifically, the results are extended to the number of weakly connected subdigraphs in a digraph based only on a subset of the first k NNs, and in a marked or labeled digraph where each vertex also has a mark or a label associated with it, and also to the number of subgraphs of the underlying k NN graphs. These constructs derived from k NN digraphs, k NN graphs, and the marked/labeled k NN graphs have applications in various fields such as pattern classification and spatial data analysis, and our extensions provide the theoretical basis for certain tools in these areas.

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关于随机 kNN 有向图中弱连通子图的数量

我们基于 $$\mathbb {R}^d$$ R d 中某些随机点过程的数据研究了 k 最近邻 (k NN) 有向图的弱连接子有向图的副本数。特别是，基于齐次泊松过程 (HPP) 和均匀二项式过程 (UBP) 点集泛函的渐近理论，我们提供了 k 个 NN 有向图的弱连通子图的数量的渐近行为的一般结果。作为推论，我们基于来自 HPP 和 UBP 的数据，获得了具有固定入度的顶点数量、共享 k NN 对的数量以及 k NN 有向图中自反 k NN 的数量的渐近结果。我们还提供了与 k NN 有向图有关的结果的几种扩展；进一步来说，结果扩展到仅基于前 k 个神经网络的一个子集的有向图中弱连接子有向图的数量，以及在标记或标记有向图中，其中每个顶点也有一个标记或与之关联的标签，并且还扩展到基础 k NN 图的子图数量。这些源自 k NN 有向图、k NN 图和标记/标记的 k NN 图的构造在模式分类和空间数据分析等各个领域都有应用，我们的扩展为这些领域的某些工具提供了理论基础。