funsPDomNum2PE1D: The functions for probability of domination number = 2 for...

In pcds: Proximity Catch Digraphs and Their Applications

The functions for probability of domination number = 2 for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

The function Pdom.num2PE1D and its auxiliary functions.

Returns P(\gamma=2) for PE-PCD whose vertices are a uniform data set of size n in a finite interval (a,b) where \gamma stands for the domination number.

The PE proximity region N_{PE}(x,r,c) is defined with respect to (a,b) with centrality parameter c \in (0,1) and expansion parameter r \ge 1.

To compute the probability P(\gamma=2) for PE-PCD in the 1D case, we partition the domain (r,c)=(1,\infty) \times (0,1), and compute the probability for each partition set. The sample size (i.e., number of vertices or data points) is a positive integer, n.

Usage

Pdom.num2AI(r, c, n)

Pdom.num2AII(r, c, n)

Pdom.num2AIII(r, c, n)

Pdom.num2AIV(r, c, n)

Pdom.num2A(r, c, n)

Pdom.num2Asym(r, c, n)

Pdom.num2BIII(r, c, n)

Pdom.num2B(r, c, n)

Pdom.num2Bsym(r, c, n)

Pdom.num2CIV(r, c, n)

Pdom.num2C(r, c, n)

Pdom.num2Csym(r, c, n)

Pdom.num2PE1D(r, c, n)

Arguments

r	A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.
c	A positive real number in (0,1) parameterizing the center inside int=(a,b). For the interval, (a,b), the parameterized center is M_c=a+c(b-a).
n	A positive integer representing the size of the uniform data set.

Value

P(domination number\le 1) for PE-PCD whose vertices are a uniform data set of size n in a finite interval (a,b)

Auxiliary Functions for Pdom.num2PE1D

The auxiliary functions are Pdom.num2AI, Pdom.num2AII, Pdom.num2AIII, Pdom.num2AIV, Pdom.num2A, Pdom.num2Asym, Pdom.num2BIII, Pdom.num2B, Pdom.num2B, Pdom.num2Bsym, Pdom.num2CIV, Pdom.num2C, and Pdom.num2Csym, each corresponding to a partition of the domain of r and c. In particular, the domain partition is handled in 3 cases as

CASE A: c \in ((3-\sqrt{5})/2, 1/2)

CASE B: c \in (1/4,(3-\sqrt{5})/2) and

CASE C: c \in (0,1/4).

Case A - c \in ((3-\sqrt{5})/2, 1/2)

In Case A, we compute P(\gamma=2) with

Pdom.num2AIV(r,c,n) if 1 < r < (1-c)/c;

Pdom.num2AIII(r,c,n) if (1-c)/c< r < 1/(1-c);

Pdom.num2AII(r,c,n) if 1/(1-c)< r < 1/c;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2A(r,c,n) combines these functions in Case A: c \in ((3-\sqrt{5})/2,1/2). Due to the symmetry in the PE proximity regions, we use Pdom.num2Asym(r,c,n) for c in (1/2,(\sqrt{5}-1)/2) with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if 1 < r < c/(1-c);

Pdom.num2AIII(r,1-c,n) if (c/(1-c) < r < 1/c;

Pdom.num2AII(r,1-c,n) if 1/c < r < 1/(1-c);

and Pdom.num2AI(r,1-c,n) otherwise.

Case B - c \in (1/4,(3-\sqrt{5})/2)

In Case B, we compute P(\gamma=2) with

Pdom.num2AIV(r,c,n) if 1 < r < 1/(1-c);

Pdom.num2BIII(r,c,n) if 1/(1-c) < r < (1-c)/c;

Pdom.num2AII(r,c,n) if (1-c)/c < r < 1/c;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2B(r,c,n) combines these functions in Case B: c \in (1/4,(3-\sqrt{5})/2). Due to the symmetry in the PE proximity regions, we use Pdom.num2Bsym(r,c,n) for c in ((\sqrt{5}-1)/2,3/4) with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if 1< r < 1/c;

Pdom.num2BIII(r,1-c,n) if 1/c < r < c/(1-c);

Pdom.num2AII(r,1-c,n) if c/(1-c) < r < 1/(1-c);

and Pdom.num2AI(r,1-c,n) otherwise.

Case C - c \in (0,1/4)

In Case C, we compute P(\gamma=2) with

Pdom.num2AIV(r,c,n) if 1< r < 1/(1-c);

Pdom.num2BIII(r,c,n) if 1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c);

Pdom.num2CIV(r,c,n) if (1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c);

Pdom.num2BIII(r,c,n) if (1+\sqrt{1-4 c})/(2 c) < r <1/(1-c);

Pdom.num2AII(r,c,n) if 1/(1-c) < r < 1/c;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2C(r,c,n) combines these functions in Case C: c \in (0,1/4). Due to the symmetry in the PE proximity regions, we use Pdom.num2Csym(r,c,n) for c \in (3/4,1) with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if 1< r < 1/c;

Pdom.num2BIII(r,1-c,n) if 1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c));

Pdom.num2CIV(r,1-c,n) if (1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c));

Pdom.num2BIII(r,1-c,n) if (1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c);

Pdom.num2AII(r,1-c,n) if c/(1-c)< r < 1/(1-c);

and Pdom.num2AI(r,1-c,n) otherwise.

Combining Cases A, B, and C, we get our main function Pdom.num2PE1D which computes P(\gamma=2) for any (r,c) in its domain.

Author(s)

Elvan Ceyhan

See Also

Pdom.num2PEtri and Pdom.num2PE1Dasy

Examples

#Examples for the main function Pdom.num2PE1D
r<-2
c<-.5

Pdom.num2PE1D(r,c,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)

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