generalized dominance relations.

positional_dominance(A, type = "one-mode", map = FALSE, benefit = TRUE)

Arguments

A

Matrix containing attributes or relations, for instance calculated by indirect_relations.

type

A string which is either 'one-mode' (Default) if A is a regular one-mode network or 'two-mode' if A is a general data matrix.

map

Logical scalar, whether rows can be sorted or not (Default). See Details.

benefit

Logical scalar, whether the attributes or relations are benefit or cost variables.

Value

Dominance relations as matrix object. An entry [u,v] is 1 if u is dominated by v.

Details

Positional dominance is a generalization of neighborhood-inclusion for arbitrary network data. In the default case, it checks for all pairs \(u,v\) if \(A_{ut} \ge A_{vt}\) holds for all \(t\) if benefit = TRUE or \(A_{ut} \le A_{vt}\) holds for all \(t\) if benefit = FALSE. This form of dominance is referred to as dominance under total heterogeneity. If map=TRUE, the rows of \(A\) are sorted decreasingly (benefit = TRUE) or increasingly (benefit = FALSE) and then the dominance condition is checked. This second form of dominance is referred to as dominance under total homogeneity, while the first is called dominance under total heterogeneity.

References

Brandes, U., 2016. Network positions. Methodological Innovations 9, 2059799116630650.

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

See also

neighborhood_inclusion, indirect_relations, exact_rank_prob

Examples

library(igraph) g <- graph.empty(n=11,directed = FALSE) g <- add_edges(g,c(1,11,2,4,3,5,3,11,4,8,5,9,5,11,6,7,6,8, 6,10,6,11,7,9,7,10,7,11,8,9,8,10,9,10)) P<-neighborhood_inclusion(g) comparable_pairs(P)
#> [1] 0.1636
# positional dominance under total heterogeneity dist <- indirect_relations(g,type = "dist_sp") D <- positional_dominance(dist,map = FALSE,benefit = FALSE) comparable_pairs(D)
#> [1] 0.1636
# positional dominance under total homogeneity D_map <- positional_dominance(dist,map = TRUE,benefit = FALSE) comparable_pairs(D_map) #more comparables than D
#> [1] 0.8727