This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

BetweennessCentrality

BetweennessCentrality[g]
gives a list of betweenness centralities for the vertices in the graph g.
  • 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 be zero when there is no path from to .
Betweenness centrality for each vertex in a connected undirected graph:
A disconnected undirected graph:
Betweenness centrality for each vertex in a directed graph:
The middle edge acts as a bridge between two strongly connected components:
There is no connection between the two components:
Betweenness centrality for each vertex in a connected undirected graph:
In[1]:=
Click for copyable input
Out[1]=
A disconnected undirected graph:
In[2]:=
Click for copyable input
Out[2]=
 
Betweenness centrality for each vertex in a directed graph:
In[1]:=
Click for copyable input
Out[1]=
The middle edge acts as a bridge between two strongly connected components:
In[2]:=
Click for copyable input
Out[2]=
There is no connection between the two components:
In[3]:=
Click for copyable input
Out[3]=
BetweennessCentrality for undirected graphs:
Highlight the vertex with highest centrality:
Directed graphs:
Highlight the vertex with highest centrality:
Works with large graphs:
Highlight the betweenness centrality for CycleGraph:
An unbalanced tree:
New in 8