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NETWORK>CENTRALITY>HUBBEL/KATZ (INFLUENCE)

PURPOSE Calculate the influence measure between every pair of vertices using the models of Hubbell, Katz or Taylor.

DESCRIPTION Successive powers of matrices provide measures of influence since they enumerate the number of possible walks of given length between all pairs of nodes. Since longer walks are assumed to contribute less in terms of influence, an attenuation factor is included and the sum of all walks is taken. Hubbell includes the identity matrix in the series whereas Katz does not.
    
For Hubbell the influence matrix is I + S(bA)^i that equals inverse of (I - bA) under certain conditions. It follows that for Katz the influence matrix is inverse of (I - bA) -I under the same condition. Taylor's measure is a normalized version of the Katz measure. For each power in the series subtract the column marginals from the row marginals and normalize by the total number of walks of that length.
 

PARAMETERS
Input dataset:
Name of file containing network to be analyzed. Data type: Valued graph.

Computational Method: 
Choices are:
   
Hubbel - influence matrix defined by inverse of (I - bA) where A is the adjacency matrix and b is the attenuation factor.

Katz - influence matrix defined by inverse of (I - bA) - I where it is the adjacency matrix and b is the attenuation factor.

Taylor - takes the Katz influence matrix and takes the column marginals from the row marginals and normalizes.

Attenuation Factor (Beta): (Default = 0.5)
The value of the attenuation factor.  This value should be smaller than the reciprocal of the absolute value of the dominant eigenvalue.  This can be guaranteed by using the simple bound that all eigenvalues are smaller than the largest row (or column) sum.

Divide matrix by overall sum: (Default = NO)
Dividing the initial matrix by the sum of all its elements guarantees that the series will converge.

Output dataset:(Default = 'Influence')
Name of file which will contain the influence matrix.  Row i column j will give actor i's influence over actor j.


LOG FILE Influence matrix.

TIMING O(N^3).

COMMENTS None.

REFERENCES Hubbell C H (1965).  'An input-output approach to clique identification'.  Sociometry, 28, 377-399.

Katz L (1953).  'A new status index derived from sociometric data analysis'.  psychometrika, 18, 34-43.

Taylor M (1969).  'Influence structures'.  Sociometry 32, 490-502.