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NETWORK > CENTRALITY > CLOSENESS

PURPOSE Calculates the farness and normalized closeness centrality  and variants of each vertex and gives the overall network closeness centralization.

DESCRIPTION The farness of a vertex is the sum of the lengths of the geodesics to every other vertex.  The reciprocal of farness is closeness centrality.  The normalized closeness centrality of a vertex is the reciprocal of farness divided by the minimum possible farness expressed as a percentage. As an alternative to taking the reciprocal after the summation, the reciprocals can be taken before. In this case the closeness is the sum of the reciprocated distances so that infinite distances contribute a value of zero. This can also be normalized by dividing by the maximum value. The reversed distance is the diameter of the graph minus the geodesic distance. In addition the routine also allows the use user to measure distance by the sums of the lengths of all the paths or all the trails. If the data is directed the routine calculates separate measures for in-closeness and out-closeness.
 
For a given network with vertices v1....vn and maximum closeness centrality cmax, the network closeness centralization measure is S(cmax - c(vi)) divided by the maximum value possible, where c(vi) is the closeness centrality of vertex vi.

   
The routine calculates centrality, network closeness centralization and some descriptive statistics based on these measures for symmetric and directed graphs.

PARAMETERS
  Input dataset:
Name of file containing network to be analyzed. Data type: Digraph
 
Type:
Choices are:
Sum of geodesic distances(Freeman) distances are lengths of shortest paths, the standard Freeman measure.
Sum of reciprocal distances distances are the reciprocal of the lengths of the geodesic paths.
Avg of reversed distances (Valente-Foreman) the reversed distance is the diameter minus the geodesic distance.
All paths distances between actors are the sums of the distances on all paths connecting them.
All trails distances between the actors are the sums of the distances on all trails connecting them.

Output Dataset: (Default = 'Closeness')
Name of file which will contain farness and normalized closeness centrality of each vertex.


LOG FILE A table which contains a list of the farness (or closeness) and normalized closeness 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.  If the data is directed then in-closeness and out-closeness are calculated seperately. Note that in-closeness for the reversed closeness is called integration and the out-closeness is called radiality.
Descriptive statistics which give the mean, standard deviation, variance minimum value and maximum value for both lists.  This is followed by the closeness network centralization index expressed as a percentage. If the data is directed then separate in and out values are calculated.

TIMING O(N^3) for Freeman and reciprocal distances, the other two can be exponential.

COMMENTS Closeness centrality be thought of as an index of the expected time-until-arrival for things flowing through the network via optimal paths. 

REFERENCES Freeman L C (1979).  'Centrality in Social Networks: Conceptual clarification'.  Social Networks 1, 215-239.
Valente, T.W., Foreman, R.K., (1998) Integration and radiality: Measuring the extent of an individual's connectedness and reachability in a network. Social Networks 20, 89-109.