alpha_centrality: Find Bonacich alpha centrality scores of network positions

In igraph: Network Analysis and Visualization

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,
  weights = NULL,
  tol = 1e-07,
  sparse = TRUE
)

Arguments

graph	The input graph, can be directed or undirected. In undirected graphs, edges are treated as if they were reciprocal directed ones.
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 centrality 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 necessarily 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 [e-mail]

References

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

See Also

eigen_centrality() and power_centrality()

Centrality measures authority_score(), betweenness(), closeness(), diversity(), eigen_centrality(), harmonic_centrality(), hits_scores(), page_rank(), power_centrality(), spectrum(), strength(), subgraph_centrality()

Examples

# The examples from Bonacich's paper
g.1 <- make_graph(c(1, 3, 2, 3, 3, 4, 4, 5))
g.2 <- make_graph(c(2, 1, 3, 1, 4, 1, 5, 1))
g.3 <- make_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)

igraph documentation built on Oct. 20, 2024, 1:06 a.m.