NETWORK > CENTRALITY > BONACICH POWER
PURPOSE Compute Bonacich's power based centrality measure for every vertex and give an overall network centralization index for this centrality measure.
DESCRIPTION Given an adjacency matrix A, the centrality of vertex i (denoted ci), is given by ci =SAij(a+bcj) where a and b are parameters. The centrality of each vertex is therefore determined by the centrality of the vertices it is connected to.
The value of a is used to normalize the measure, the value of b is an attenuation factor which gives the amount of dependence of each vertex's centrality on the centralities of the vertices it is adjacent to. The normalization parameter is automatically selected so that the square root of the sum of squares of the vertex centralities is the size of the network. (That is the Euclidean norm of the vector equals the number of vertices).
The parameter b is selected by the user, negative values should be selected if an individual's power is increased by being connected to vertices with low power and positive values selected if an individual's power is increased by being connected to vertices with high power. Note a value of zero would give the out-degree of each vertex.
The routine calculates power centrality and some descriptive statistics of the measure. To methods are given one method is exact but this can be slow for large networks. The other is an iterative routine which will give the same normalized answer if allowed to run for a long time in all but a few unusual situations.
Name of file containing network to be analyzed. Data type: Valued Digraph.
Output Power dataset: (Default = 'BonacichPower- Exact/Iterative').
Name of file which contains power centrality measure for every vertex. The extension depends on the method chosen.
Exact implements a standard eigenvector type method which can be computationally expensive for large networks but should be the preferred method.
Iterative gives a power method calculation. This can sometimes not converge regardless of the walk length, although this is rare.
Value of attenuation factor (Beta): (Default = 0.0).
A value of 0 gives a centrality measure directly proportional to the out-degree of each vertex. Positive values give weight to being connected to powerful actors, negative values give weight to being connected to low powered actors. Larger values in modulus gives greater weight to actors further away. This parameter must be smaller in modulus than the reciprocal of the largest eigenvalue. To find the highest positive value then clicking on the Get Beta button calculates this value. It enters a value for beta that is 0.5% lower than the maximum possible. Note this needs to be repeated for every new dataset entered.
Maximum Walk Length
The iterative method proceeds by taking a starting vector and iteratively calculating the centrality scores and normalizing at each iteration. If this was continued to convergence then it would produce the same answer as the exact method. For large matrices it is better to stop this after a fixed number of steps and these would correspond to the walk lengths in the digraph. This methods fails when there is not a single dominat eigenvalue.
LOG FILE A table which contains the power centrality of each actor.
TIMING O(N^3) for the exact method
REFERENCES Bonacich P (1987). Power and Centrality: A family of Measures. American Journal of Sociology 92, 1170-1182.