This vignette describes the concept of positional dominance, the generalization of neighborhood-inclusion for arbitrary network and attribute data. Additionally, some use cases with the `netrankr`

package are given. The partial ranking induced by positional dominance can be used to assess partial centrality or compute probabilistic centrality.

A network can be described as a *dyadic variable* \(x\in \mathcal{W}^\mathcal{D}\), where \(\mathcal{W}\) is the value range of the network (in the simple case of unweighted networks \(\mathcal{W}=\{0,1\}\)) and \(\mathcal{D}=\mathcal{N}\times\mathcal{A}\) describes the dyadic domain of actors \(\mathcal{N}\) and affiliations \(\mathcal{A}\). If \(\mathcal{A}\neq\mathcal{N}\), we obtain a *two-mode network* and if \(\mathcal{A}=\mathcal{N}\) a *one-mode network* consisting of relations among actors.

**Definition**

Let \(x\in \mathcal{W}^\mathcal{D}\) be a network and \(i,j \in \mathcal{N}\). We say that \(i\) is dominated by \(j\) *under the total homogeneity assumption*, denoted by \(i \leq j\) if \[
x_{it}\leq x_{jt} \quad \forall t \in \mathcal{N}.
\] If there exists a permutation \(\pi: \mathcal{N} \to \mathcal{N}\) such that \[
x_{it}\leq x_{j\pi(t)} \quad \forall t \in \mathcal{N},
\] we say that \(i\) is dominated by \(j\) *under the total heterogeneity assumption*, denoted by \(i ⪯ j\).

It holds that \(i\leq j \implies i ⪯ j\) but not vice versa.

More about the positional dominance and the positional approach to network analysis can be found in

Brandes, Ulrik. (2016). Network Positions.

Methodological Innovations,9, 2059799116630650. (link)

`netrankr`

Package```
library(netrankr)
library(igraph)
library(magrittr)
set.seed(1886) #for reproducibility
```

The function `positional_dominance`

can be used to check both, dominance under homogeneity and heterogeneity. In accordance with the analytic pipeline of centrality (consult the tutorial REFREF) we use the `%>%`

operator from the `magrittr`

package.

```
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))
V(g)$name <- 1:11
#neighborhood inclusion can be expressed with the analytic pipeline
D <- g %>% indirect_relations(type="identity") %>% positional_dominance()
#without %>% operator:
# rel <- indirect_relations(g,type="identity")
# D <- positional_dominance(rel)
#check if identical to neighborhood-inclusion
identical(D,neighborhood_inclusion(g))
```

`## [1] TRUE`

More on the `indirect_relations()`

function can be found in this vignette.

The `map`

parameter of `positional_dominance`

allows to distinguish between dominance under *total heterogeneity* (`map=FALSE`

) and *total homogeneity* (`map=TRUE`

). In the later case, all relations can be ordered non-decreasingly (or non-increasingly if the relation describes costs, such as distances) and afterwards checked front to back. Dominance under total homogeneity yields a ranking, if the relation is binary (e.g. adjacent or not).

```
D <- g %>%
indirect_relations(type="identity") %>%
positional_dominance(map=TRUE)
comparable_pairs(D)
```

`## [1] 1`

For cost variables like shortes path distances, the `benefit`

parameter is set to `FALSE`

.

```
D1 <- g %>%
indirect_relations(type="dist_sp") %>%
positional_dominance(map=FALSE,benefit=FALSE)
```

From the definition given in the first section, it is clear that there are always at least as many comparable pairs under the total homogeneity assumption as under total heterogeneity.

```
D1 <- g %>%
indirect_relations(type="dist_sp") %>%
positional_dominance(map=FALSE,benefit=FALSE)
D2 <- g %>%
indirect_relations(type="dist_sp") %>%
positional_dominance(map=TRUE,benefit=FALSE)
c("heterogeneity"=comparable_pairs(D1),
"homogeneity"=comparable_pairs(D2))
```

```
## heterogeneity homogeneity
## 0.1636364 0.8727273
```

Additionally, all dominance relations from the heterogeneity assumption are preserved under total homogeneity. (Note: \(A\implies B\) is equivalent to \(\neg A \lor B\))

`all(D1!=1 | D2==1) `

`## [1] TRUE`