Contents - Index

NETWORK > CENTRALITY > BETWEENNESS > NODES

PURPOSE Calculates the betweenness and normalized betweenness centrality of each vertex and gives the overall network betweenness centralization.

DESCRIPTION Let bjk be the proportion of all geodesics linking vertex j and vertex k which pass through vertex i.  The betweenness of vertex i is the sum of all bjk where i, j and k are distinct.  Betweenness is therefore a measure of the number of times a vertex occurs on a geodesic.  The normalized betweenness centrality is the betweenness divided by the maximum possible betweenness expressed as a percentage.  

For a given network with vertices v1....vn and maximum betweenness centrality cmax, the network betweenness centralization measure is S(cmax - c(vi)) divided by the maximum value possible, where c(vi) is the betweenness centrality of vertex vi.

The routine calculates these measures, and some descriptive statistics based on these measures, for symmetric and unsymmetric graphs.

PARAMETERS
Input dataset:
Name of file containing network to be analyzed. Data type: Digraph.
 
 
Output dataset: (Default = 'FreemanBetweenness').
Name of file which will contain betweenness and normalized betweenness centrality of each vertex.


LOG FILE A table which contains a list of the betweenness and normalized betweenness centrality expressed as a percentage for each vertex.These have been ordered so that the actor with the highest centrality appears first. Note the stored UCINET output file retains the original order.  
  
Descriptive statistics which give the mean, standard deviation, variance, minimum value and maximum value for both lists.  This is followed by the betweenness network centralization index expressed as a percentage.

TIMING O(N^3).

COMMENTS Betweenness centrality measures information control.  
Care should be taken in interpreting betweenness for directed data.

REFERENCES Freeman L C (1979).  'Centrality in Social Networks: Conceptual Clarification'.  Social Networks 1, 215-239.