RadialityCentrality
gives a list of radiality centralities for the vertices in the graph g.
RadialityCentrality[g,"In"]
gives a list of in-centralities for a directed graph g.
RadialityCentrality[g,"Out"]
gives a list of out-centralities for a directed graph g.
RadialityCentrality[{vw,…},…]
uses rules vw to specify the graph g.
Details
- Radiality in-centralities are also known as integration centralities.
- RadialityCentrality will give high centralities to vertices that are a short distance to every other vertex in its reachable neighborhood compared to its diameter.
- Radiality out-centrality for a vertex
is computed using the out component 
for the vertex 
and is given by 
, where 
is the distance from 
to 
in 
, 
is the diameter of 
, and the sum is over the vertices in 
. - Radiality in-centrality for a vertex
is computed using the in component 
for the vertex 
and is given by 
, where 
is the distance from 
to 
in 
, 
is the diameter of 
, and the sum is over the vertices in 
. - The radiality centrality for an isolated vertex is taken to be zero.
- RadialityCentrality works with undirected graphs, directed graphs, weighted graphs, multigraphs, and mixed graphs.
Examples
open all close allBasic Examples (2)
Compute radiality centralities:
g = ExampleData[{"NetworkGraph", "Friendship"}];RadialityCentrality[g]HighlightGraph[g, VertexList[g], VertexSize -> Thread[VertexList[g] -> Rescale[%]]]Rank vertices. Highest-ranked vertices are at a short distance to all reachable vertices compared to the highest distance in the graph:
g = ExampleData[{"NetworkGraph", "Friendship"}];Part[VertexList[g], Ordering[RadialityCentrality[g], All, Greater]]Scope (8)
RadialityCentrality works with undirected graphs:
RadialityCentrality[[image]]RadialityCentrality[[image]]RadialityCentrality[[image]]RadialityCentrality[[image]]RadialityCentrality[[image]]Use rules to specify the graph:
RadialityCentrality[{1 -> 3, 2 -> 1, 3 -> 6, 4 -> 6, 1 -> 5, 5 -> 4, 6 -> 1}]Compute in-centralities and out-centralities:
RadialityCentrality[[image], "In"]RadialityCentrality[[image], "Out"]RadialityCentrality works with large graphs:
g = RandomGraph[{10000, 10005}];RadialityCentrality[g]//Short//TimingApplications (2)
Rank vertices of a graph by the degree of easiness to reach other vertices:
g = [image];SortBy[{VertexList[g], RadialityCentrality[g]}, Last]//ReverseHighlight the radiality centrality for CycleGraph:
HighlightCentrality[g_, cc_] := HighlightGraph[g, Table[Style[VertexList[g][[i]], ColorData["TemperatureMap"][cc[[i]] / Max[cc]]], {i, VertexCount[g]}]];g = CycleGraph[8, VertexSize -> Large];cc = RadialityCentrality[g];HighlightCentrality[g, cc]g = GridGraph[{10, 10}, VertexSize -> Large];cc = RadialityCentrality[g];HighlightCentrality[g, cc]g = CompleteKaryTree[3, 3, VertexSize -> Large];cc = RadialityCentrality[g];HighlightCentrality[g, cc]g = PathGraph[Range[20], VertexSize -> Large];cc = RadialityCentrality[g];HighlightCentrality[g, cc]Properties & Relations (3)
Radiality centrality is between 0 and 1:
Through[{Min, Max}[RadialityCentrality [ RandomGraph[{100, 200}]]]]RadialityCentrality can be computed using GraphDistanceMatrix:
g = [image];GraphDiameter[g] + 1 - GraphDistanceMatrix[g] //N// MatrixFormTable[Mean[Drop[%[[i]], {i}]] / GraphDiameter[g], {i, VertexCount[g]}]RadialityCentrality[g] == %Use VertexIndex to obtain the centrality of a specific vertex:
g = ExampleData[{"NetworkGraph", "Friendship"}];RadialityCentrality[g][[VertexIndex[g, "Anna"]]]Text
Wolfram Research (2012), RadialityCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/RadialityCentrality.html (updated 2015).
CMS
Wolfram Language. 2012. "RadialityCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/RadialityCentrality.html.
APA
Wolfram Language. (2012). RadialityCentrality. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RadialityCentrality.html