Network Centrality and Power
centpow infilename , [ bonacich normalize beta(real) symmetrize(upper | lower | sum) saveas(outfilename)]
Description
centpow creates a new dataset containing measures of centrality and power for each node in a network described by infilename. This command does not affect the data in memory.
Options bonacich Computes eigenvector centrality, and beta centrality and power. normalize Computes normalized alter-based centrality and power in binary networks. beta(real) Specifies the value of the beta parameter used to compute beta centrality and power (must be positive). The default is set just below the maximum allowable value (.99 * 1/largest eigenvalue). symmetrize Transforms an asymmetric matrix into a symmetric matrix using the specified method. The default method is sum. saveas(outfilename) Specifies the name of the output file containing the centrality and power scores. The default is centpow.dta.
Symmetrize Suboptions Degree centrality, alter-based centrality, and alter-based power are only applicable to undirected networks and thus can be computed only for symmetric matrices. When infilename is asymmetric, the symmetrize option can be used to transform it into a symmetric matrix. Three transformations are available:
upper - Use only the upper half; Replace Rji with Rij, i < j lower - Use only the lower half; Replace Rij with Rji, i < j sum - Sum the upper and lower halves; Replace both Rij and Rji with (Rij + Rji) , i < j
Input outfilename must be a comma-delimited file containing a square matrix. It may be symmetric or asymmetric, and its entries may be binary or valued. Entries on the diagonal are ignored.
Output Each vector of scores is saved as a variable in outfilename.dta, with the observations in the same order as the rows and columns of the matrix in infilename.
Degree centrality is defined as a node's total number of connections, and is saved as the variable degree.
Alter-based centrality and power are extensions of degree centrality in which each node's score is based on the degree centrality of the nodes to which it is connected. Alter-based centrality is defined as the sum of the degree centralities of a node's alters, and is saved as the variable altercent. Alter-based power is defined as the sum of the inverse degree centralities of a node's alters, and is saved as the variable alterpow. Optional normalization rescales these variables to range between 0 (the minimum possible value in a network of the given size) and 1 (the maximum possible value); normalized values are saved as the variables n_altercent and n_alterpow.
Eigenvector centrality is saved as the variable eigencent, while beta centrality and power are saved as the variables betacent and betapow, respectively. These measures are conceptually similar to alter-based centrality and power, and often yield similar results. However, they are computationally distinct (see Bonacich 1987) and are subject to several assumptions:
(1) The network contains no disconnected components,
(2) The largest eigenvalue is substantially larger than the 2nd largest eigenvalue,
(3) For beta centrality/power: The absolute value of beta is less than the reciprocal of the largest eigenvalue.
If any of these assumptions is violated, the command will issue a warning, but will still compute the measures.
References
Neal, Z. P. (in press) Differentiating Centrality and Power in the World City Network, Urban Studies.
Neal, J. W. and Z. P. Neal. (in press) Power as a Structural Phenomenon, American Journal of Community Psychology.
Bonacich, P. (1987) Power and Centrality: A Family of Measures, American Journal of Sociology 92: 1070-82.
Author
Zachary Neal Department of Sociology Michigan State University zpneal@msu.edu