{smcl}
{* 12jan2011}{...}
{hline}
help for {hi:centpow}
{hline}

{title:Network Centrality and Power}

{p 8 16 2}{cmd:centpow}
   {it:infilename}
   {cmd:,}
     [ {cmd:{ul:bon}acich}
     {cmd:{ul:norm}alize}
     {cmd:beta(}{it:real}{cmd:)}
     {cmd:{ul:sym}metrize(}upper | lower | sum{cmd:)}
     {cmd:saveas(}{it:outfilename}{cmd:)}]


{title:Description}

{p 4 4 2}
{cmd:centpow} creates a new dataset containing measures of centrality and power for each node in a network described by {it:infilename}.
This command does not affect the data in memory.


{title:Options}
{p 4 8 2}{cmd:bonacich} Computes eigenvector centrality, and beta centrality and power.{p_end}
{p 4 8 2}{cmd:normalize} Computes normalized alter-based centrality and power in binary networks.{p_end}
{p 4 8 2}{cmd:beta(}{it:real}{cmd:)} Specifies the value of the {it: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).{p_end}
{p 4 8 2}{cmd:symmetrize} Transforms an asymmetric matrix into a symmetric matrix using the specified method.  The default method is {it:sum}.{p_end}
{p 4 8 2}{cmd:saveas(}{it:outfilename}{cmd:)} Specifies the name of the output file containing the centrality and power scores.  The default is {it:centpow.dta}.{p_end}


{title:Symmetrize Suboptions}
{p 4 4 2}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 {it:infilename} is asymmetric, the {cmd:symmetrize} option can be used to transform it into a symmetric matrix.
Three transformations are available:{p_end}

{p 4 4 2}{it:upper} - Use only the upper half; Replace Rji with Rij, i < j{p_end}
{p 4 4 2}{it:lower} - Use only the lower half; Replace Rij with Rji, i < j{p_end}
{p 4 4 2}{it:sum} - Sum the upper and lower halves; Replace both Rij and Rji with (Rij + Rji) , i < j{p_end}


{title:Input}
{p 4 4 2}{it: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.{p_end}


{title:Output}
{p 4 4 2}Each vector of scores is saved as a variable in {it:outfilename}.dta, with the observations in the same order as the rows and columns of the matrix in {it:infilename}.

{p 4 4 2}Degree centrality is defined as a node's total number of connections, and is saved as the variable {it:degree}.

{p 4 4 2}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 {it: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 {it: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 {it:n_altercent} and {it:n_alterpow}.

{p 4 4 2}Eigenvector centrality is saved as the variable {it:eigencent}, while beta centrality and power are saved as the variables {it:betacent} and {it: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:{p_end}

{p 8 8 2}(1) The network contains no disconnected components,{p_end}

{p 8 8 2}(2) The largest eigenvalue is substantially larger than the 2nd largest eigenvalue,{p_end}

{p 8 8 2}(3) For beta centrality/power: The absolute value of {it:beta} is less than the reciprocal of the largest eigenvalue.{p_end}

{p 8 8 2}If any of these assumptions is violated, the command will issue a warning, but will still compute the measures.{p_end}


{title:References}

{p 0 5}
Neal, Z. P. (in press)  Differentiating Centrality and Power in the World City Network, {it:Urban Studies}.

{p 0 5}
Neal, J. W. and Z. P. Neal.  (in press)  Power as a Structural Phenomenon, {it:American Journal of Community Psychology}.

{p 0 5}
Bonacich, P.  (1987)  Power and Centrality: A Family of Measures, {it:American Journal of Sociology} 92: 1070-82.


{title:Author}

Zachary Neal
Department of Sociology
Michigan State University
zpneal@msu.edu