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


  nodes = V(graph),
  alpha = 1,
  loops = FALSE,
  exo = 1,
  weights = NULL,
  tol = 1e-07,
  sparse = TRUE



The input graph, can be directed or undirected. In undirected graphs, edges are treated as if they were reciprocal directed ones.


Vertex sequence, the vertices for which the alpha centrality values are returned. (For technical reasons they will be calculated for all vertices, anyway.)


Parameter specifying the relative importance of endogenous versus exogenous factors in the determination of centrality. See details below.


Whether to eliminate loop edges from the graph before the calculation.


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.


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.


Tolerance for near-singularities during matrix inversion, see solve().


Logical scalar, whether to use sparse matrices for the calculation. The ‘Matrix’ package is required for sparse matrix support


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


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.


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


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


Gabor Csardi


# 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))
#> [1] 1 1 3 4 5
#> [1] 5 1 1 1 1
alpha_centrality(g.3, alpha = 0.5)
#> [1] 2.533333 2.266667 2.133333 2.066667 1.000000