Metrika

, Volume 75, Issue 6, pp 761–793 | Cite as

The distribution of the relative arc density of a family of interval catch digraph based on uniform data

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Abstract

We study a family of interval catch digraph called proportional-edge proximity catch digraph (PCD) which is also a special type of intersection digraphs parameterized with an expansion and a centrality parameter. PCDs are random catch digraphs that have been developed recently and have applications in classification and spatial pattern analysis. We investigate a graph invariant of the PCDs called relative arc density. We demonstrate that relative arc density of PCDs is a U-statistic and using the central limit theory of U-statistics, we derive the (asymptotic) distribution of the relative arc density of proportional-edge PCD for uniform data in one dimension. We also determine the parameters for which the rate of convergence to asymptotic normality is fastest.

Keywords

Class cover catch digraph Intersection digraph Proximity catch digraph Proximity map Random graph U-statistics 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsKoç UniversitySarıyer, IstanbulTurkey

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