netrankr (version 1.2.4)

neighborhood_inclusion: Neighborhood-inclusion preorder

Description

Calculates the neighborhood-inclusion preorder of an undirected graph.

Usage

neighborhood_inclusion(g, sparse = FALSE)

Value

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

Arguments

g

An igraph object

sparse

Logical scalar, whether to create a sparse matrix

Author

David Schoch

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., Müller, 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

library(igraph)
# the neighborhood inclusion preorder of a star graph is complete
g <- make_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))