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.
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**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.
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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`

Packagelibrary(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 we use the `%>%`

operator.

data("dbces11") g <- dbces11 #neighborhood inclusion can be expressed with the analytic pipeline D <- g %>% indirect_relations(type="adjacency") %>% positional_dominance()

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="adjacency") %>% positional_dominance(map=TRUE) comparable_pairs(D)

For cost variables like shortest 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))

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)

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