bonpow takes a graph stack (dat) and returns the Bonacich power centralities of positions within one graph (indicated by nodes and g, respectively). 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=c(1:dim(dat)[2]), gmode="digraph", diag=FALSE,
tmaxdev=FALSE, exponent=1, rescale=FALSE, tol=1e-07)g=1.Diag is FALSE by default.tmaxdev=FALSE.solve)bonpow will not symmetrize your data before extracting eigenvectors; don't send this routine asymmetric matrices unless you really mean to do so.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. (1987). ``Power and Centrality: A Family of Measures.'' American Journal of Sociology, 92, 1170-1182.
centralization, evcent#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)Run the code above in your browser using DataLab