gives a list of betweenness centralities for the vertices in the graph g.
uses rules vw to specify the graph g.
- 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.
Examplesopen allclose all
Basic Examples (2)
BetweennessCentrality works with undirected graphs:
Use rules to specify the graph:
BetweennessCentrality works with large graphs:
Rank vertices by the fraction of shortest paths between other vertices:
Highlight the betweenness centrality for CycleGraph:
Find the members who can easily withhold or distort information in transmission in a 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:
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:
Wolfram Research (2010), BetweennessCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/BetweennessCentrality.html (updated 2015).
Wolfram Language. 2010. "BetweennessCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/BetweennessCentrality.html.
Wolfram Language. (2010). BetweennessCentrality. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetweennessCentrality.html