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On the Number of Weakly Connected Subdigraphs in Random kNN Digraphs

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Abstract

We study the number of copies of a weakly connected subdigraph of the k nearest neighbor (kNN) digraph based on data from certain random point processes in \(\mathbb {R}^d\). 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 kNN digraphs. As corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared kNN pairs, and the number of reflexive kNNs in the kNN digraph based on data from HPP and UBP. We also provide several extensions of our results pertaining to the kNN 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 kNN graphs. These constructs derived from kNN digraphs, kNN graphs, and the marked/labeled kNN 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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Authors and Affiliations

  1. Department of Mathematics, Ankara Yıldırım Beyazıt University, Ankara, Turkey

    Selim Bahadır

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Elvan Ceyhan

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  1. Selim Bahadır
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  2. Elvan Ceyhan
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Correspondence to Elvan Ceyhan.

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Bahadır, S., Ceyhan, E. On the Number of Weakly Connected Subdigraphs in Random kNN Digraphs. Discrete Comput Geom 65, 116–142 (2021). https://doi.org/10.1007/s00454-020-00218-8

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  • DOI: https://doi.org/10.1007/s00454-020-00218-8

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