Pdom.num2PEtri: Asymptotic probability that domination number of Proportional...

In pcds: Proximity Catch Digraphs and Their Applications

Asymptotic probability that domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) equals 2 where vertices of the digraph are uniform points in a triangle

Description

Returns P(domination number=2) for PE-PCD for uniform data in a triangle, when the sample size n goes to infinity (i.e., asymptotic probability of domination number = 2).

PE proximity regions are constructed with respect to the triangle with the expansion parameter r \ge 1 and M-vertex regions where M is the vertex that renders the asymptotic distribution of the domination number non-degenerate for the given value of r in (1,1.5].

See also (\insertCiteceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011;textualpcds).

Usage

Pdom.num2PEtri(r)

Arguments

r	A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,1.5] to attain non-degenerate asymptotic distribution for the domination number.

Value

P(domination number=2) for PE-PCD for uniform data on an triangle as the sample size n goes to infinity

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

Pdom.num2PE1D

Examples

Pdom.num2PEtri(r=1.5)
Pdom.num2PEtri(r=1.4999999999)

Pdom.num2PEtri(r=1.5) / Pdom.num2PEtri(r=1.4999999999)

rseq<-seq(1.01,1.49999999999,l=20)  #try also l=100
lrseq<-length(rseq)

pg2<-vector()
for (i in 1:lrseq)
{
  pg2<-c(pg2,Pdom.num2PEtri(rseq[i]))
}

plot(rseq, pg2,type="l",xlab="r",
ylab=expression(paste("P(", gamma, "=2)")),
     lty=1,xlim=range(rseq)+c(0,.01),ylim=c(0,1))
points(rbind(c(1.50,Pdom.num2PEtri(1.50))),pch=".",cex=3)

pcds documentation built on May 29, 2024, 9:36 a.m.