Closeness centrality — closeness

Closeness centrality of Freeman (1978) and its harmonic version (Marchiori and Latora, 2000; Rochat, 2009).

Usage

closeness_centrality(
  A,
  digraph = TRUE,
  type = c("out", "in", "all"),
  weighted = FALSE,
  alpha = 1,
  harmonic = FALSE,
  normalized = FALSE
)

Arguments

A: A square matrix
digraph: Whether the matrix is directed or undirected
type: Whether to use the out (default), in or all distances. The all option uses the underlying graph
weighted: Whether the matrix is weighted
alpha: The tuning parameter of Opsahl et al. (2010) to transform weights into lengths
harmonic: Whether to return the harmonic closeness
normalized: If TRUE, Freeman's closeness is multiplied by the number of nodes reached, and the harmonic closeness is divided by (n-1)

Value

This function returns the closeness centrality of the nodes.

Details

Freeman's closeness is the inverse of the sum of the geodesic distances from a node to the others. When the network is disconnected, the sum only considers the nodes that can be reached, and the harmonic version is recommended, as it adds the inverse of each distance (an unreachable node adds zero).

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

Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215–239. doi:10.1016/0378-8733(78)90021-7

Marchiori, M. and Latora, V. (2000). Harmony in the small-world. Physica A, 285(3-4), 539–546. doi:10.1016/S0378-4371(00)00311-3

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

Rochat, Y. (2009). Closeness centrality extended to unconnected graphs: The harmonic centrality index. ASNA.

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

closeness_centrality(A, digraph = FALSE)
#>         a         b         c         d         e 
#> 0.2000000 0.1250000 0.1666667 0.1250000 0.1111111 
closeness_centrality(A, digraph = FALSE, harmonic = TRUE)
#>        a        b        c        d        e 
#> 3.500000 2.333333 3.000000 2.333333 2.166667