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ClosenessCentrality

ClosenessCentrality[g]
finds the closeness centrality.
  • The closeness centrality of a vertex u is defined as the inverse of the sum of the distance from u to all other vertices. The closeness centrality of a vertex in a disconnected graph is based on the closeness centrality of the component where this vertex belongs.
  • The following options can be given:
WeightedTruewhether edge weight is to be used in calculating distance
NormalizeFalsewhether to normalize the output
This defines a small graph:
This shows that vertex 2, being at the center of this graph, has a higher closeness centrality:
This defines a disconnected graph and finds the closeness centrality:
Needs["GraphUtilities`"]
This defines a small graph:
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Out[3]=
This shows that vertex 2, being at the center of this graph, has a higher closeness centrality:
In[4]:=
Click for copyable input
Out[4]=
 
Needs["GraphUtilities`"]
This defines a disconnected graph and finds the closeness centrality:
In[2]:=
Click for copyable input
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
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Out[4]=
The centrality of a vertex that cannot reach all other vertices in its component is zero:
The centrality of a disconnected graph is calculated by treating each component separately:
This defines a graph with edge weights:
By default, edge weights are taken into account:
This gives the closeness centrality if edge weights are assumed to be 1.:
A plot of a grid graph with vertices of high centrality in red: