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NETWORK > CENTRALITY > EIGENVECTOR

PURPOSE Calculates the eigenvector of the largest positive eigenvalue as a measure of centrality.

DESCRIPTION Given an adjacency matrix A, the centrality of vertex i (denoted ci), is given by  ci =aSAijcj where a is a parameter.  The centrality of each vertex is therefore determined by the centrality of the vertices it is connected to. The parameter ? is required to give the equations a non-trivial solution and is therefore the reciprocal of an eigenvalue.  It follows that the centralities will be the elements of the corresponding eigenvector.  The normalized eigenvector centrality is the scaled eigenvector centrality divided by the maximum difference possible expressed as a percentage.

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

This routine calculates these measures and some descriptive statistics based on these measures. This routine only handles symmetric data and in these circumstances the eigenvalues provide a measure of the accuracy of the centrality measure.  To help interpretation the routine calculates all positive eigenvalues but only gives the eigenvector corresponding to the largest eigenvalue.

PARAMETERS
Input dataset:
Name of file containing network to be analyzed. Data type: Valued Graph (Symmetric data only).

Output dataset: (Default = 'BonacichCentrality').
Name of file which will contain eigenvector centrality measure for each vertex.

LOG FILE A table of positive eigenvalues.  The eigenvalues are placed in descending order under the heading VALUE.  The table gives information on 'how dominant' the largest eigenvalue is.  The table gives the percentage and cumulative percentage of the total eigenvalue sum for each eigenvalue.  The ratio of each eigenvalue to the next largest is also presented.

This is followed by a list of vertices which contains the eigenvector and normalized eigenvector centrality measure for every vertex.  These values should be interpreted in terms of an interval scale.

Finally the network eigenvector centralization index expressed as a percentage is given.

TIMING O(N^3).

COMMENTS The ratio of the largest eigenvalue to the next largest should be at least 1.5 and preferably 2.0 or more for the centrality measure to be robust.  If this is not the case then a full factor analysis should be undertaken.

REFERENCES Bonacich P (1972).  Factoring and Weighting Approaches to status scores and clique identification.  Journal of Mathematical Sociology 2, 113-120.