# neighborhood_inclusion: Neighborhood-inclusion preorder In netrankr: Analyzing Partial Rankings in Networks

## Description

Calculates the neighborhood-inclusion preorder of an undirected graph.

## Usage

 `1` ```neighborhood_inclusion(g, sparse = FALSE) ```

## Arguments

 `g` An igraph object `sparse` Logical scalar, whether to create a sparse matrix

## Details

Neighborhood-inclusion is defined as

N(u)\subseteq N[v]

where N(u) is the neighborhood of u and N[v]=N(v)\cup \lbrace v\rbrace is the closed neighborhood of v. N(u) \subseteq N[v] implies that c(u) ≤q c(v), where c is a centrality index based on a specific path algebra. Indices falling into this category are closeness (and variants), betweenness (and variants) as well as many walk-based indices (eigenvector and subgraph centrality, total communicability,...).

## Value

The neighborhood-inclusion preorder of `g` as matrix object. `P[u,v]=1` if N(u)\subseteq N[v]

David Schoch

## References

Schoch, D. and Brandes, U., 2016. Re-conceptualizing centrality in social networks. European Journal of Applied Mathematics 27(6), 971-985.

Brandes, U. Heine, M., Müller, J. and Ortmann, M., 2017. Positional Dominance: Concepts and Algorithms. Conference on Algorithms and Discrete Applied Mathematics, 60-71.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```library(igraph) # the neighborhood inclusion preorder of a star graph is complete g <- graph.star(5, "undirected") P <- neighborhood_inclusion(g) comparable_pairs(P) # the same holds for threshold graphs tg <- threshold_graph(50, 0.1) P <- neighborhood_inclusion(tg) comparable_pairs(P) # standard centrality indices preserve neighborhood-inclusion data("dbces11") P <- neighborhood_inclusion(dbces11) is_preserved(P, degree(dbces11)) is_preserved(P, closeness(dbces11)) is_preserved(P, betweenness(dbces11)) ```