# alpha_centrality: Find Bonacich alpha centrality scores of network positions In igraph: Network Analysis and Visualization

## Description

`alpha_centrality` calculates the alpha centrality of some (or all) vertices in a graph.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```alpha_centrality( graph, nodes = V(graph), alpha = 1, loops = FALSE, exo = 1, weights = NULL, tol = 1e-07, sparse = TRUE ) ```

## Arguments

 `graph` The input graph, can be directed or undirected `nodes` Vertex sequence, the vertices for which the alpha centrality values are returned. (For technical reasons they will be calculated for all vertices, anyway.) `alpha` Parameter specifying the relative importance of endogenous versus exogenous factors in the determination of centrality. See details below. `loops` Whether to eliminate loop edges from the graph before the calculation. `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. `weights` A character scalar that gives the name of the edge attribute to use in the adjacency matrix. If it is `NULL`, then the ‘weight’ edge attribute of the graph is used, if there is one. Otherwise, or if it is `NA`, then the calculation uses the standard adjacency matrix. `tol` Tolerance for near-singularities during matrix inversion, see `solve`. `sparse` Logical scalar, whether to use sparse matrices for the calculation. The ‘Matrix’ package is required for sparse matrix support

## 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:

x=alpha t(A)x+e,

where A is the (not neccessarily symmetric) adjacency matrix of the graph, e is the vector of exogenous sources of status of the vertices and alpha is the relative importance of the endogenous versus exogenous factors.

## Value

A numeric vector contaning the centrality scores for the selected vertices.

## Warning

Singular adjacency matrices cause problems for this algorithm, the routine may fail is certain cases.

## Author(s)

Gabor Csardi csardi.gabor@gmail.com

## References

Bonacich, P. and Lloyd, P. (2001). “Eigenvector-like measures of centrality for asymmetric relations” Social Networks, 23, 191-201.

`eigen_centrality` and `power_centrality`
 ```1 2 3 4 5 6 7``` ```# 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) ```