Rank intervals

The ranks that a node can take in the rankings that are consistent with a dominance relation (Schoch and Brandes, 2016).

Usage

dominance_ranks(P, direction = c("dominated", "dominates"))

Arguments

P

A binary dominance matrix, such as the output of neigh_inclusion()

direction

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

Value

This function returns the minimum and the maximum rank of every node, and the width of the interval.

Details

A dominance relation only orders some pairs of nodes. Any centrality index that preserves it gives a complete ranking, but different indices give different rankings. The interval of a node contains every rank it can take in such a ranking: it cannot be ranked below the nodes it dominates, nor above the nodes that dominate it. A node with a wide interval is one whose position depends on the index that is chosen, and a node with an interval of a single value has the same rank under every index that preserves the relation.

The rank one is the lowest.

References

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)

dominance_ranks(neigh_inclusion(A))
#>   node min_rank max_rank width
#> 1    a        4        5     1
#> 2    b        1        2     1
#> 3    c        4        5     1
#> 4    d        1        2     1
#> 5    e        1        3     2