generalized dominance relations that can be computed on one and two mode networks.

## Usage

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.

David Schoch

## Examples

library(igraph)

data("dbces11")

P <- neighborhood_inclusion(dbces11)
comparable_pairs(P)
#> [1] 0.1636364

# positional dominance under total heterogeneity
dist <- indirect_relations(dbces11, type = "dist_sp")
D <- positional_dominance(dist, map = FALSE, benefit = FALSE)
comparable_pairs(D)
#> [1] 0.1636364

# positional dominance under total homogeneity
D_map <- positional_dominance(dist, map = TRUE, benefit = FALSE)
comparable_pairs(D_map)
#> [1] 0.8727273