Calculates the neighborhood-inclusion preorder of an undirected graph.

neighborhood_inclusion(g)

## Arguments

g An igraph object

## Value

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

## 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 $$c(u) \leq c(v)$$, 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,...).

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.

## Examples

library(igraph)
#the neighborhood inclusion preorder of a star graph is complete
g <- graph.star(5,'undirected')
P <- neighborhood_inclusion(g)
comparable_pairs(P)#>  1
#the same holds for threshold graphs
tg <- threshold_graph(50,0.1)
P <- neighborhood_inclusion(tg)
comparable_pairs(P)#>  1
#standard centrality indices preserve neighborhood-inclusion
g <- graph.empty(n=11,directed = FALSE)
is_preserved(P,degree(g))#>  TRUEis_preserved(P,closeness(g))#>  TRUEis_preserved(P,betweenness(g))#>  TRUE