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Neighbourhood inclusion in directed networks

Source: R/dominance.R
dir_inclusion.Rd

Neighbourhood-inclusion preorders for directed networks (Marmulla and Brandes, 2026), which extend the vicinal preorder of the undirected case to the criteria that different families of centrality indices preserve.

Usage

dir_inclusion(
  A,
  type = c("radial_out", "radial_in", "hierarchical_down", "hierarchical_up", "medial"),
  strength = c("strong", "weak"),
  direction = c("dominated", "dominates")
)

Arguments

A

A square matrix

type

The criterion: radial_out (default), radial_in, hierarchical_down, hierarchical_up or medial

strength

Whether the neighbourhoods are open (strong, default) or closed (weak). It is ignored for the medial criterion

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 criterion asked for in type, where P[u, v] = 1 when the neighbourhoods of u are included in those of v, in the direction asked for.

Details

Let \(N^+(i)\) be the nodes that \(i\) sends ties to, \(N^-(i)\) the ones that send ties to \(i\), and \(N[i]\) the same set including \(i\). The strong relations use the open neighbourhoods and the weak ones the closed neighbourhoods:

radial_out: \(N^+(i) \subseteq N^+(j)\), preserved by the indices that measure how far a node reaches, such as out-degree and closeness.

radial_in: \(N^-(i) \subseteq N^-(j)\), the same for the ties received.

hierarchical_down: \(N^+(i) \subseteq N^+(j)\) and \(N^-(i) \supseteq N^-(j)\), so \(j\) sends more and receives less than \(i\).

hierarchical_up: \(N^-(i) \subseteq N^-(j)\) and \(N^+(i) \supseteq N^+(j)\), so \(j\) receives more and sends less, which is the criterion of the indices of status.

medial: \(N^+(i) \subseteq N^+[j]\) and \(N^-(i) \subseteq N^-[j]\), with two extra conditions when \(i\) and \(j\) are adjacent, so that the advantage that the dominated node has from that tie is compensated. It is the criterion preserved by betweenness.

References

Marmulla, G. and Brandes, U. (2026). Centrality in directed networks. Social Networks, 86, 23–34. doi:10.1016/j.socnet.2026.01.001

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

dir_inclusion(A, type = "radial_out")
#>   a b c d e
#> a 0 0 0 0 0
#> b 1 0 0 0 0
#> c 0 0 0 0 0
#> d 0 0 1 0 0
#> e 1 1 1 1 0
dir_inclusion(A, type = "medial")
#>   a b c d e
#> a 0 1 0 0 0
#> b 0 0 0 0 0
#> c 0 0 0 0 0
#> d 0 0 0 0 0
#> e 0 0 0 1 0