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% Generated by roxygen2 (4.1.1): do not edit by hand
% Please edit documentation in R/structural.properties.R
\name{alpha_centrality}
\alias{alpha.centrality}
\alias{alpha_centrality}
\title{Find Bonacich alpha centrality scores of network positions}
\usage{
alpha_centrality(graph, nodes = V(graph), alpha = 1, loops = FALSE,
exo = 1, weights = NULL, tol = 1e-07, sparse = TRUE)
}
\arguments{
\item{graph}{The input graph, can be directed or undirected}
\item{nodes}{Vertex sequence, the vertices for which the alpha centrality
values are returned. (For technical reasons they will be calculated for all
vertices, anyway.)}
\item{alpha}{Parameter specifying the relative importance of endogenous
versus exogenous factors in the determination of centrality. See details
below.}
\item{loops}{Whether to eliminate loop edges from the graph before the
calculation.}
\item{exo}{The exogenous factors, in most cases this is either a constant --
the same factor for every node, or a vector giving the factor for every
vertex. Note that too long vectors will be truncated and too short vectors
will be replicated to match the number of vertices.}
\item{weights}{A character scalar that gives the name of the edge attribute
to use in the adjacency matrix. If it is \code{NULL}, then the
\sQuote{weight} edge attribute of the graph is used, if there is one.
Otherwise, or if it is \code{NA}, then the calculation uses the standard
adjacency matrix.}
\item{tol}{Tolerance for near-singularities during matrix inversion, see
\code{\link{solve}}.}
\item{sparse}{Logical scalar, whether to use sparse matrices for the
calculation. The \sQuote{Matrix} package is required for sparse matrix
support}
}
\value{
A numeric vector contaning the centrality scores for the selected
vertices.
}
\description{
\code{alpha_centrality} calculates the alpha centrality of some (or all)
vertices in a graph.
}
\details{
The alpha centrality measure can be considered as a generalization of
eigenvector centerality to directed graphs. It was proposed by Bonacich in
2001 (see reference below).
The alpha centrality of the vertices in a graph is defined as the solution
of the following matrix equation: \deqn{x=\alpha A^T x+e,}{x=alpha t(A)x+e,}
where \eqn{A}{A} is the (not neccessarily symmetric) adjacency matrix of the
graph, \eqn{e}{e} is the vector of exogenous sources of status of the
vertices and \eqn{\alpha}{alpha} is the relative importance of the
endogenous versus exogenous factors.
}
\section{Warning}{
Singular adjacency matrices cause problems for this
algorithm, the routine may fail is certain cases.
}
\examples{
# The examples from Bonacich's paper
g.1 <- graph( c(1,3,2,3,3,4,4,5) )
g.2 <- graph( c(2,1,3,1,4,1,5,1) )
g.3 <- graph( c(1,2,2,3,3,4,4,1,5,1) )
alpha_centrality(g.1)
alpha_centrality(g.2)
alpha_centrality(g.3,alpha=0.5)
}
\author{
Gabor Csardi \email{csardi.gabor@gmail.com}
}
\references{
Bonacich, P. and Paulette, L. (2001). ``Eigenvector-like
measures of centrality for asymmetric relations'' \emph{Social Networks},
23, 191-201.
}
\seealso{
\code{\link{eigen_centrality}} and \code{\link{power_centrality}}
}
\keyword{graphs}
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