Betweenness centrality — betweenness

Betweenness centrality of Freeman (1977), computed with the algorithm of Brandes (2001).

Usage

betweenness_centrality(
  A,
  digraph = TRUE,
  weighted = FALSE,
  alpha = 1,
  normalized = FALSE
)

Arguments

A: A square matrix
digraph: Whether the matrix is directed or undirected
weighted: Whether the matrix is weighted
alpha: The tuning parameter of Opsahl et al. (2010) to transform weights into lengths
normalized: If TRUE, the result is divided by (n-1)(n-2) for directed networks and (n-1)(n-2)/2 for undirected networks

Value

This function returns the betweenness centrality of the nodes.

Details

The betweenness of a node is the sum, over all pairs of other nodes, of the proportion of geodesics between the pair that pass through the node. For undirected networks each pair is counted once.

For valued matrices, the tie weights are treated as strengths and transformed into lengths as \(1 / w^{\alpha}\) (Opsahl et al., 2010). If alpha = 0 the binary network is used.

References

Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25(2), 163–177. doi:10.1080/0022250X.2001.9990249

Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40(1), 35–41. doi:10.2307/3033543

Opsahl, T., Agneessens, F., and Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks, 32(3), 245–251. doi:10.1016/j.socnet.2010.03.006

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 1, 0, 0, 0, 0, 0,
  1, 0, 1, 1, 1, 0, 0, 0, 0,
  1, 1, 0, 1, 0, 1, 0, 0, 0,
  1, 1, 1, 0, 1, 1, 0, 0, 0,
  0, 1, 0, 1, 0, 1, 1, 0, 0,
  0, 0, 1, 1, 1, 0, 1, 0, 0,
  0, 0, 0, 0, 1, 1, 0, 1, 0,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0
), byrow = TRUE, ncol = 9)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

betweenness_centrality(A, digraph = FALSE)
#>         a         b         c         d         e         f         g         h 
#>  0.000000  1.583333  1.583333  3.166667  6.333333  6.333333 12.000000  7.000000 
#>         i 
#>  0.000000