Neighbourhood inclusion

Neighbourhood-inclusion preorder of an undirected network (Brandes, 2016; Schoch and Brandes, 2016).

Usage

neigh_inclusion(A, closed = TRUE, direction = c("dominated", "dominates"))

Arguments

A

A symmetric matrix object

closed

Whether the neighbourhood of the dominating node is closed (i.e. includes the node itself)

direction

Whether P[u, v] = 1 means that u is dominated by v (default, as in Schoch and Brandes, 2016) or that u dominates v

Value

This function returns a binary matrix P of the neighbourhood inclusion, where P[u, v] = 1 when the neighbours of u are also neighbours of v, in the direction asked for.

Details

A node \(u\) is dominated by a node \(v\) if the open neighbourhood of \(u\) is included in the closed neighbourhood of \(v\), \(N(u) \subseteq N[v]\). With closed = FALSE both neighbourhoods are open, \(N(u) \subseteq N(v)\).

In matrix form, \(u\) is dominated by \(v\) when the number of shared neighbours equals the degree of \(u\).

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, 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)

neigh_inclusion(A)
#>   a b c d e f g h i
#> a 0 1 1 1 0 0 0 0 0
#> b 0 0 0 1 0 0 0 0 0
#> c 0 0 0 1 0 0 0 0 0
#> d 0 0 0 0 0 0 0 0 0
#> e 0 0 0 0 0 0 0 0 0
#> f 0 0 0 0 0 0 0 0 0
#> g 0 0 0 0 0 0 0 0 0
#> h 0 0 0 0 0 0 0 0 0
#> i 0 0 0 0 0 0 1 1 0