`bonpow`

takes one or more graphs (`dat`

) and returns the Boncich power centralities of positions (selected by `nodes`

) within the graphs indicated by `g`

. The decay rate for power contributions is specified by `exponent`

(1 by default). This function is compatible with `centralization`

, and will return the theoretical maximum absolute deviation (from maximum) conditional on size (which is used by `centralization`

to normalize the observed centralization score).

```
bonpow(dat, g=1, nodes=NULL, gmode="digraph", diag=FALSE,
tmaxdev=FALSE, exponent=1, rescale=FALSE, tol=1e-07)
```

dat

one or more input graphs.

g

integer indicating the index of the graph for which centralities are to be calculated (or a vector thereof). By default, `g`

=1.

nodes

vector indicating which nodes are to be included in the calculation. By default, all nodes are included.

gmode

string indicating the type of graph being evaluated. `"digraph"`

indicates that edges should be interpreted as directed; `"graph"`

indicates that edges are undirected. This is currently ignored.

diag

boolean indicating whether or not the diagonal should be treated as valid data. Set this true if and only if the data can contain loops. `Diag`

is `FALSE`

by default.

tmaxdev

boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, `tmaxdev`

=`FALSE`

.

exponent

exponent (decay rate) for the Bonacich power centrality score; can be negative

rescale

if true, centrality scores are rescaled such that they sum to 1.

tol

tolerance for near-singularities during matrix inversion (see `solve`

)

A vector, matrix, or list containing the centrality scores (depending on the number and size of the input graphs).

Singular adjacency matrices cause no end of headaches for this algorithm; thus, the routine may fail in certain cases. This will be fixed when I get a better algorithm. `bonpow`

will not symmetrize your data before extracting eigenvectors; don't send this routine asymmetric matrices unless you really mean to do so.

Bonacich's power centrality measure is defined by \(C_{BP}\left(\alpha,\beta\right)=\alpha\left(\mathbf{I}-\beta\mathbf{A}\right)^{-1}\mathbf{A}\mathbf{1}\), where \(\beta\) is an attenuation parameter (set here by `exponent`

) and \(\mathbf{A}\) is the graph adjacency matrix. (The coefficient \(\alpha\) acts as a scaling parameter, and is set here (following Bonacich (1987)) such that the sum of squared scores is equal to the number of vertices. This allows 1 to be used as a reference value for the ``middle'' of the centrality range.) When \(\beta \rightarrow 1/\lambda_{\mathbf{A}1}\) (the reciprocal of the largest eigenvalue of \(\mathbf{A}\)), this is to within a constant multiple of the familiar eigenvector centrality score; for other values of \(\beta\), the behavior of the measure is quite different. In particular, \(\beta\) gives positive and negative weight to even and odd walks, respectively, as can be seen from the series expansion \(C_{BP}\left(\alpha,\beta\right)=\alpha \sum_{k=0}^\infty \beta^k \mathbf{A}^{k+1} \mathbf{1}\) which converges so long as \(|\beta| < 1/\lambda_{\mathbf{A}1}\). The magnitude of \(\beta\) controls the influence of distant actors on ego's centrality score, with larger magnitudes indicating slower rates of decay. (High rates, hence, imply a greater sensitivity to edge effects.)

Interpretively, the Bonacich power measure corresponds to the notion that the power of a vertex is recursively defined by the sum of the power of its alters. The nature of the recursion involved is then controlled by the power exponent: positive values imply that vertices become more powerful as their alters become more powerful (as occurs in cooperative relations), while negative values imply that vertices become more powerful only as their alters become *weaker* (as occurs in competitive or antagonistic relations). The magnitude of the exponent indicates the tendency of the effect to decay across long walks; higher magnitudes imply slower decay. One interesting feature of this measure is its relative instability to changes in exponent magnitude (particularly in the negative case). If your theory motivates use of this measure, you should be very careful to choose a decay parameter on a non-ad hoc basis.

Bonacich, P. (1972). ``Factoring and Weighting Approaches to Status Scores and Clique Identification.'' *Journal of Mathematical Sociology*, 2, 113-120.

Bonacich, P. (1987). ``Power and Centrality: A Family of Measures.'' *American Journal of Sociology*, 92, 1170-1182.

```
# NOT RUN {
#Generate some test data
dat<-rgraph(10,mode="graph")
#Compute Bonpow scores
bonpow(dat,exponent=1,tol=1e-20)
bonpow(dat,exponent=-1,tol=1e-20)
# }
```

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