# Graph based spatial weights

### Description

Functions return a graph object containing a list with the vertex
coordinates and the to and from indices defining the edges. Some/all of these functions assume that the coordinates are not exactly regularly spaced. The helper
function `graph2nb`

converts a graph
object into a neighbour list. The plot functions plot the graph objects.

### Usage

1 2 3 4 5 6 7 8 9 | ```
gabrielneigh(coords, nnmult=3)
relativeneigh(coords, nnmult=3)
soi.graph(tri.nb, coords, quadsegs=10)
graph2nb(gob, row.names=NULL,sym=FALSE)
## S3 method for class 'Gabriel'
plot(x, show.points=FALSE, add=FALSE, linecol=par(col), ...)
## S3 method for class 'relative'
plot(x, show.points=FALSE, add=FALSE, linecol=par(col),...)
``` |

### Arguments

`coords` |
matrix of region point coordinates |

`nnmult` |
scaling factor for memory allocation, default 3; if higher values are required, the function will exit with an error; example below thanks to Dan Putler |

`tri.nb` |
a neighbor list created from tri2nb |

`quadsegs` |
number of line segments making a quarter circle buffer, see |

`gob` |
a graph object created from any of the graph funtions |

`row.names` |
character vector of region ids to be added to the
neighbours list as attribute |

`sym` |
a logical argument indicating whether or not neighbors should be symetric (if i->j then j->i) |

`x` |
object to be plotted |

`show.points` |
(logical) add points to plot |

`add` |
(logical) add to existing plot |

`linecol` |
edge plotting colour |

`...` |
further graphical parameters as in |

### Details

The graph functions produce graphs on a 2d point set that are all subgraphs of the Delaunay triangulation. The relative neighbor graph is defined by the relation, x and y are neighbors if

*d(x,y) <= min(max(d(x,z),d(y,z))| z in S)*

where d() is the distance, S is the set of points and z is an arbitrary point in S. The Gabriel graph is a subgraph of the delaunay triangulation and has the relative neighbor graph as a sub-graph. The relative neighbor graph is defined by the relation x and y are Gabriel neighbors if

*d(x,y) <= min((d(x,z)^2 + d(y,z)^2)^1/2 |z in S)*

where x,y,z and S are as before. The sphere of influence graph is
defined for a finite point set S, let *r_x* be the distance from point x
to its nearest neighbor in S, and *C_x* is the circle centered on x. Then
x and y are SOI neigbors iff *C_x* and *C_y* intersect in at
least 2 places. From 2016-05-31, Computational Geometry in C code replaced by calls to functions in RANN and rgeos; with a large `quadsegs=`

argument, the behaviour of the function is the same, otherwise buffer intersections only closely approximate the original function.

See `card`

for details of “nb” objects.

### Value

A list of class `Graph`

withte following elements

`np` |
number of input points |

`from` |
array of origin ids |

`to` |
array of destination ids |

`nedges` |
number of edges in graph |

`x` |
input x coordinates |

`y` |
input y coordinates |

The helper functions return an `nb`

object with a list of integer
vectors containing neighbour region number ids.

### Author(s)

Nicholas Lewin-Koh nikko@hailmail.net

### References

Matula, D. W. and Sokal R. R. 1980, Properties of Gabriel graphs relevant to geographic variation research and the clustering of points in the plane, Geographic Analysis, 12(3), pp. 205-222.

Toussaint, G. T. 1980, The relative neighborhood graph of a finite planar set, Pattern Recognition, 12(4), pp. 261-268.

Kirkpatrick, D. G. and Radke, J. D. 1985, A framework for computational morphology. In Computational Geometry, Ed. G. T. Toussaint, North Holland.

### See Also

`knearneigh`

, `dnearneigh`

,
`knn2nb`

, `card`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | ```
example(columbus)
coords <- coordinates(columbus)
par(mfrow=c(2,2))
col.tri.nb<-tri2nb(coords)
col.gab.nb<-graph2nb(gabrielneigh(coords), sym=TRUE)
col.rel.nb<- graph2nb(relativeneigh(coords), sym=TRUE)
plot(columbus, border="grey")
plot(col.tri.nb,coords,add=TRUE)
title(main="Delaunay Triangulation")
plot(columbus, border="grey")
plot(col.gab.nb, coords, add=TRUE)
title(main="Gabriel Graph")
plot(columbus, border="grey")
plot(col.rel.nb, coords, add=TRUE)
title(main="Relative Neighbor Graph")
plot(columbus, border="grey")
if (require(rgeos, quietly=TRUE) && require(RANN, quietly=TRUE)) {
col.soi.nb<- graph2nb(soi.graph(col.tri.nb,coords), sym=TRUE)
plot(col.soi.nb, coords, add=TRUE)
title(main="Sphere of Influence Graph")
}
par(mfrow=c(1,1))
dx <- rep(0.25*0:4,5)
dy <- c(rep(0,5),rep(0.25,5),rep(0.5,5), rep(0.75,5),rep(1,5))
m <- cbind(c(dx, dx, 3+dx, 3+dx), c(dy, 3+dy, dy, 3+dy))
try(res <- gabrielneigh(m))
res <- gabrielneigh(m, nnmult=4)
summary(graph2nb(res))
grd <- as.matrix(expand.grid(x=1:5, y=1:5)) #gridded data
r2 <- gabrielneigh(grd)
set.seed(1)
grd1 <- as.matrix(expand.grid(x=1:5, y=1:5)) + matrix(runif(50, .0001, .0006), nrow=25)
r3 <- gabrielneigh(grd1)
opar <- par(mfrow=c(1,2))
plot(r2, show=TRUE, linecol=2)
plot(r3, show=TRUE, linecol=2)
par(opar)
``` |