netrankr (version 0.2.1)

neighborhood_inclusion: Neighborhood-inclusion preorder

Description

Calculates the neighborhood-inclusion preorder of an undirected graph.

Usage

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,...).

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., M<U+00FC>ller, J. and Ortmann, M., 2017. Positional Dominance: Concepts and Algorithms. Conference on Algorithms and Discrete Applied Mathematics, 60-71.

See Also

positional_dominance, exact_rank_prob

Examples

# NOT RUN {
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
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))
P <- neighborhood_inclusion(g)

is_preserved(P,degree(g))
is_preserved(P,closeness(g))
is_preserved(P,betweenness(g))
# }