# alpha.centrality

##### Find Bonacich alpha centrality scores of network positions

`alpha.centrality`

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

- Keywords
- graphs

##### Usage

```
alpha.centrality(graph, nodes=V(graph), alpha=1, loops=FALSE,
exo=1, weights=NULL, tol=1e-7)
```

##### 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 first, 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 long vectors will be truncated and short vectors will be replicated.
- weights
- Optional positive weight vector for calculating
weighted closeness. If the graph has a
`weight`

edge attribute, then this is used by default. - tol
- Tolerance for near-singularities during matrix inversion,
see
`solve`

.

##### 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: $$x=\alpha A^T x+e,$$ 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.

##### concept

Alpha centrality

##### Warning

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

##### References

Bonacich, P. and Paulette, L. (2001). ``Eigenvector-like measures of
centrality for asymmetric relations'' *Social Networks*, 23,
191-201.

##### See Also

##### Examples

```
# The examples from Bonacich's paper
g.1 <- graph( c(1,3,2,3,3,4,4,5)-1 )
g.2 <- graph( c(2,1,3,1,4,1,5,1)-1 )
g.3 <- graph( c(1,2,2,3,3,4,4,1,5,1)-1 )
alpha.centrality(g.1)
alpha.centrality(g.2)
alpha.centrality(g.3,alpha=0.5)
```

*Documentation reproduced from package igraph, version 0.5.3, License: GPL (>= 2)*