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Positional dominance on indirect relations

Source: R/dominance.R
pos_dominance.Rd

Dominance between the rows of a matrix of relations, which do not need to be the ties themselves (Brandes, 2016; Schoch and Brandes, 2016).

Usage

pos_dominance(
  R,
  map = FALSE,
  benefit = TRUE,
  direction = c("dominated", "dominates")
)

Arguments

R

A square matrix of relations, such as the output of indirect_rel()

map

Whether the values are sorted before being compared (total homogeneity)

benefit

Whether a larger value is better

direction

Whether P[u, v] = 1 means that u is dominated by v (default) or that u dominates v

Value

This function returns a binary matrix P of the positional dominance, where P[u, v] = 1 when every indirect relation of u is at most the one of v, in the direction asked for.

Details

Neighbourhood inclusion compares the ties of the nodes. The same comparison can be made on any relation derived from the network, such as the distances between the nodes or the number of walks that join them, which is what makes different centrality indices comparable.

Under total heterogeneity (map = FALSE) the values are compared one by one: \(i\) is dominated by \(j\) when its relation with every other node is at most as large. Under total homogeneity (map = TRUE) the values are sorted before being compared, so it does not matter with whom the relation is held, only how large the values are.

With benefit = FALSE a smaller value is better, which is the case of distances.

References

Brandes, U. (2016). Network positions. Methodological Innovations, 9, 1–19. doi:10.1177/2059799116630650

Schoch, D. and Brandes, U. (2016). Re-conceptualizing centrality in social networks. European Journal of Applied Mathematics, 27(6), 971–985. doi:10.1017/S0956792516000401

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)

# Distances: the closer to the others, the better
D <- indirect_rel(A, type = "distance", digraph = FALSE)
pos_dominance(D, benefit = FALSE)
#>   a b c d e
#> a 0 0 0 0 0
#> b 1 0 1 1 0
#> c 0 0 0 0 0
#> d 1 1 1 0 0
#> e 1 0 1 0 0
pos_dominance(D, benefit = FALSE, map = TRUE)
#>   a b c d e
#> a 0 0 0 0 0
#> b 1 0 1 1 0
#> c 1 0 0 0 0
#> d 1 1 1 0 0
#> e 1 1 1 1 0