SphereOfInfluenceGraph - Maple Help

GraphTheory[GeometricGraphs]

 SphereOfInfluenceGraph
 construct a sphere of influence graph

 Calling Sequence SphereOfInfluenceGraph( P, opts )

Parameters

 P - Matrix or list of lists representing set of points opts - (optional) one or more options as specified below

Options

 • vertices : list of integers, strings or symbols
 Specifies the vertices to be used in the generated graph.
 • weighted : true or false
 If weighted=true, the result is a weighted graph whose edge weights correspond to the distance between points using the specified norm. Default is false.

Description

 • The SphereOfInfluenceGraph(P, opts) command returns the sphere of influence graph for the point set P.
 • The parameter P must be a Matrix or list of lists representing a set of points.

Definition

 • Given a set $P$ of points, center a ball $B\left(p\right)$ at each point $p$ with radius equal to the distance between $p$ and its nearest neighbor in $P$.
 • The sphere of influence graph is the graph whose vertices correspond to points in $P$ and in which an edge between $p$ and $q$ exists if $B\left(p\right)\cap B\left(q\right)$ contains more than one point.
 • The sphere of influence graph is not guaranteed to be connected.
 • The undirected nearest neighbor graph is a subgraph of the sphere of influence graph.

Examples

Generate a set of random two-dimensional points amd draw a sphere of influence graph.

 > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
 > $\mathrm{with}\left(\mathrm{GeometricGraphs}\right):$
 > $\mathrm{points}≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(60,2,\mathrm{generator}=0..100.,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)$
 ${\mathrm{points}}{≔}\begin{array}{c}\left[\begin{array}{cc}{9.85017697341803}& {82.9750304386195}\\ {86.0670183749663}& {83.3188659363996}\\ {64.3746795546741}& {73.8671607639673}\\ {57.3670557294666}& {2.34399775883031}\\ {23.6234264844933}& {52.6873367387328}\\ {47.0027547350003}& {22.2459488367552}\\ {74.9213491558963}& {62.0471820220718}\\ {92.1513434709073}& {96.3107262637080}\\ {48.2319624355944}& {63.7563267144141}\\ {90.9441877431805}& {33.8527464913022}\\ {⋮}& {⋮}\end{array}\right]\\ \hfill {\text{60 × 2 Matrix}}\end{array}$ (1)
 > $\mathrm{SOIG}≔\mathrm{SphereOfInfluenceGraph}\left(\mathrm{points}\right)$
 ${\mathrm{SOIG}}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 60 vertices and 97 edge\left(s\right)}}$ (2)
 > $\mathrm{DrawGraph}\left(\mathrm{SOIG}\right)$

Compatibility

 • The GraphTheory[GeometricGraphs][SphereOfInfluenceGraph] command was introduced in Maple 2020.