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.


open allclose all

Basic Examples  (2)

Compute betweenness centralities:


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

Scope  (6)

BetweennessCentrality works with undirected graphs:

Directed graphs:


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:




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:

Wolfram Research (2010), BetweennessCentrality, Wolfram Language function, (updated 2015).


Wolfram Research (2010), BetweennessCentrality, Wolfram Language function, (updated 2015).


Wolfram Language. 2010. "BetweennessCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015.


Wolfram Language. (2010). BetweennessCentrality. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_betweennesscentrality, author="Wolfram Research", title="{BetweennessCentrality}", year="2015", howpublished="\url{}", note=[Accessed: 22-July-2024 ]}


@online{reference.wolfram_2024_betweennesscentrality, organization={Wolfram Research}, title={BetweennessCentrality}, year={2015}, url={}, note=[Accessed: 22-July-2024 ]}