BetweennessCentrality

BetweennessCentrality[g]

gives a list of betweenness centralities for the vertices in the graph g.

BetweennessCentrality[{vw,}]

uses rules vw to specify the graph g.

Details

  • BetweennessCentrality will give high centralities to vertices that are on many shortest paths of other vertex pairs.
  • BetweennessCentrality for a vertex in a connected graph is given by , where is the number of shortest paths from to and is the number of shortest paths from to passing through .
  • The ratio is taken to be zero when there is no path from to .
  • BetweennessCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (2)

Compute betweenness centralities:

Highlight:

Rank vertices. Highest-ranked vertices are on many shortest paths of other vertex pairs:

Scope  (6)

BetweennessCentrality works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

BetweennessCentrality works with large graphs:

Applications  (6)

Rank vertices by the fraction of shortest paths between other vertices:

Highlight the betweenness centrality for CycleGraph:

GridGraph:

CompleteKaryTree:

PathGraph:

Find the members who can easily withhold or distort information in transmission in a network:

Highlight the network:

A power grid network representing the topology of the Western States Power Grid of the United States. Identify critical nodes whose failures will most affect the grid:

The neighborhood of the critical nodes:

Find the toll station that would collect the most money in a toll road network:

For graphs with vertices, the largest sum in differences in betweenness centrality between the most central vertex and all other vertices is :

Measure how central the most central vertex is with respect to other vertices:

Centralization of social networks:

Properties & Relations  (3)

Betweenness centralities for an undirected graph are equivalent to centralities for each component:

Computing the centralities for each component yields the same result:

Betweenness centralities for isolated vertices are taken to be zero:

Use VertexIndex to obtain the centrality of a specific vertex:

Introduced in 2010
 (8.0)
 |
Updated in 2014
 (10.0)
2015
 (10.3)