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Dominance layers

Source: R/dominance.R
dominance_layers.Rd

Layers, status and transitive reduction of a strict dominance relation.

Usage

dominance_layers(D, reduction = FALSE, direction = c("dominated", "dominates"))

Arguments

D

A binary dominance matrix without cycles, such as the output of pareto_dominance()

reduction

Whether to return the transitive reduction of D

direction

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

Value

This function returns the layers, the layer, status and net dominance of each node and, if requested, the transitive reduction (with the same direction as D).

Details

The first layer contains the nodes that are not dominated by any other node (maximal elements). These nodes are removed, and the procedure is repeated until all nodes are assigned. Each layer is an antichain, i.e. the nodes within a layer do not dominate each other.

The maximal elements are of two kinds (Espinosa-Rada, 2026): dominant nodes dominate at least one other node, while independent nodes neither dominate nor are dominated. The remaining nodes are dominated. The net dominance is the number of nodes dominated minus the number of nodes dominating.

The transitive reduction removes the tie \(u \to v\) when there is another node \(k\) such that \(u \to k \to v\), which is the usual representation of a hierarchy (Hasse diagram).

References

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

Espinosa-Rada, A. (2026). Network positions within scholars and intellectual networks. Journal of Informetrics, 20, 101854. doi:10.1016/j.joi.2026.101854

Author

Alejandro Espinosa-Rada

Examples

X <- matrix(c(
  1, 1, 1, 0,
  1, 1, 0, 0,
  1, 0, 0, 0,
  0, 1, 0, 0,
  0, 0, 1, 1
), byrow = TRUE, ncol = 4)
rownames(X) <- c("a1", "a2", "a3", "a4", "a5")
colnames(X) <- c("w1", "w2", "w3", "w4")

D <- pareto_dominance(list(set_inclusion(X)))
dominance_layers(D, reduction = TRUE)
#> $layers
#> $layers[[1]]
#> [1] "a1" "a5"
#> 
#> $layers[[2]]
#> [1] "a2"
#> 
#> $layers[[3]]
#> [1] "a3" "a4"
#> 
#> 
#> $layer_id
#> a1 a2 a3 a4 a5 
#>  1  2  3  3  1 
#> 
#> $status
#>            a1            a2            a3            a4            a5 
#>    "dominant"   "dominated"   "dominated"   "dominated" "independent" 
#> 
#> $net_dominance
#> a1 a2 a3 a4 a5 
#>  3  1 -2 -2  0 
#> 
#> $reduction
#>    a1 a2 a3 a4 a5
#> a1  0  0  0  0  0
#> a2  1  0  0  0  0
#> a3  0  1  0  0  0
#> a4  0  1  0  0  0
#> a5  0  0  0  0  0
#>