Distribution of components in the \(k\)-nearest neighbour random geometric graph for \(k\) below the connectivity threshold. (English) Zbl 1288.60128

Falgas-Ravry, Victor

Electron. J. Probab. 18, Paper No. 83, 22 p. (2013).

Consider a Poisson point process of intensity 1 in the plane. A random geometric graph \(G\) is defined on the set \(V\) of points of the process inside a square of area \(n\) by joining each point in \(V\) to its \(k\)-nearest neighbours in \(V\). The distribution of small connected components of \(G\) is studied for \(k=k(n)\) below the connectivity threshold. It is also shown that such components are in a specified sense not close together.

Reviewer: Ove Frank (Stockholm)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
05C80	Random graphs (graph-theoretic aspects)

Keywords:

Random geometric graphs; Poisson process; nearest neighbour, connected components