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NETWORK > CENTRALITY > FLOW BETWEENNESS

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

DESCRIPTION Let mjk be the amount of flow between vertex j and vertex k which must pass through i for any maximum flow.  The flow betweenness of vertex i is the sum of all mjk where i, j and k are distinct and j < k.  The flow betweenness is therefore a measure of the contribution of a vertex to all possible maximum flows.  

The normalized flow betweenness centrality of a vertex i is the flow betweenness of i divided by the total flow through all pairs of points where i is not a source or sink.
    
For a given binary network with vertices v1....vn and maximum flow betweenness centrality cmax, the network flow betweenness centralization measure is S(cmax - c(vi)) divided by the maximum value possible, where c(vi) is the flow betweenness centrality of vertex vi.

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

PARAMETERS 
Input dataset:
Name of file containing network to be analyzed. Data type: Valued symmetric graph - integer values only.

Output dataset: (Default = 'FlowBetweenness').
Name of file which will contain flow-betweenness and normalized flow betweenness centrality of each vertex.


LOG FILE  A table which contains a list of the flow-betweenness and normalized flow betweenness (nFlowbet) centrality expressed as a percentage for each vertex.  Descriptive statistics which give the mean, standard deviation, variance, minimum value and maximum value for both lists.  

This is followed by the flow betweenness network centralization index expressed as a percentage.

TIMING O(N^4).

COMMENTS The measure is based upon the concept of information flow.  In valued data the values should in some way correspond to the capacity for flow, hence valued data should represent similarity.

REFERENCES Freeman L C, Borgatti S P and White D R (1991).  'Centrality in valued graphs: A measure of betweenness based on network flow'.  Social Networks 13, 141-154.