igraph (version 0.4.1)

alpha.centrality: Find Bonacich alpha centrality scores of network positions

Description

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

Usage

alpha.centrality(graph, nodes=V(graph), alpha=1, loops=FALSE,
                 exo=1, 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.
tol
Tolerance for near-singularities during matrix inversion, see solve.

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.

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.

References

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

See Also

evcent and bonpow

Examples

Run this code
# 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)

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