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

## 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.

## 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.

## 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
```