×

Laws of large numbers of subgraphs in directed random geometric networks. (English) Zbl 1399.05202

Summary: Given independent random points \(\mathcal X_n= \{X_1,\dots, X_n\}\) in \(\mathbb R^2\), drawn according to some probability density function \(f\) on \(\mathbb R^2\), and a cutoff \(r_n > 0\) we construct a random geometric digraph \(G(\mathcal X_n,\mathcal Y_n, r_n)\) with vertex set \(\mathcal X_n\). Each vertex \(X_i\) is assigned uniformly at random a sector \(S_i\), of central angle \(\alpha\) with inclination \(Y_i\), in a circle of radius \(r_n\) (with vertex \(X_i\) as the origin). An arc is present from \(X_i\) to \(X_j\), if \(X_j\) falls in \(S_i\). Another random geometric digraph \(G(\mathcal X_n,\mathcal R_n)\) with random radius is also introduced. In this paper we investigate two kinds of small subgraphs – induced and isolated – in the above two directed networks, which contribute to understanding the local topology of many spatial networks, such as wireless communication networks. We give some strong laws of large numbers of subgraph counts thus extending those results of M. Penrose [Random geometric graphs. Oxford: Oxford University Press (2003; Zbl 1029.60007)].

MSC:

05C80 Random graphs (graph-theoretic aspects)

Citations:

Zbl 1029.60007
PDFBibTeX XMLCite
Full Text: arXiv Link