ASedge.dens.tri {pcds.ugraph}

Edge density of the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns the edge density of the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

AS proximity regions are defined with respect to tri and vertex regions are defined with the center M="CC" for circumcenter of tri; or M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri; default is M="CC", i.e., circumcenter of tri. For the number of edges, loops are not allowed so edges are only possible for points inside tri for this function.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, edge density is computed only for the points inside the triangle (i.e., edge density of the subgraph of the underlying or reflexivity graph induced by the vertices in the triangle is computed), otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

See also (Ceyhan (2005, 2016)).

Usage

ASedge.dens.tri(
  Xp,
  tri,
  M = "CC",
  ugraph = c("underlying", "reflexivity"),
  in.tri.only = FALSE
)

Arguments

Xp

A set of 2D points which constitute the vertices of the underlying or reflexivity graph of the AS-PCD.

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC", i.e., the circumcenter of tri.

ugraph

The type of the graph based on AS-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

in.tri.only

A logical argument (default is in.tri.only=FALSE) for computing the edge density for only the points inside the triangle, tri. That is, if in.tri.only=TRUE edge density of the induced subgraph with the vertices inside tri is computed, otherwise otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

Value

Edge density of the underlying or reflexivity graphs based on the AS-PCD whose vertices are the 2D numerical data set, Xp; AS proximity regions are defined with respect to the triangle tri and M-vertex regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

PEedge.dens.tri, CSedge.dens.tri, and ASarc.dens.tri

Examples

#\donttest{
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-pcds::runif.tri(n,Tr)$g

M<-as.numeric(pcds::runif.tri(1,Tr)$g)

#For the underlying graph
(num.edgesAStri(Xp,Tr,M)$num.edges)/(n*(n-1)/2)
ASedge.dens.tri(Xp,Tr,M)
ASedge.dens.tri(Xp,Tr,M,in.tri.only = TRUE)

#For the reflexivity graph
(num.edgesAStri(Xp,Tr,M,ugraph="r")$num.edges)/(n*(n-1)/2)
ASedge.dens.tri(Xp,Tr,M,ugraph="r")
ASedge.dens.tri(Xp,Tr,M,in.tri.only = TRUE,ugraph="r")
#}