On the Number of Weakly Connected Subdigraphs in Random

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 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.