positional_dominance function

Generalized Dominance Relations

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

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.

Returns

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.

Examples

library(igraph)

data("dbces11")

P <- neighborhood_inclusion(dbces11)
comparable_pairs(P)

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

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

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

Author(s)

David Schoch