Calculates the neighborhood-inclusion preorder of an undirected graph.
neighborhood_inclusion(g, sparse = FALSE)
An igraph object
Logical scalar, whether to create a sparse matrix
The neighborhood-inclusion preorder of
g as matrix object.
P[u,v]=1 if \(N(u)\subseteq N[v]\)
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.
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 data("dbces11") P <- neighborhood_inclusion(dbces11) is_preserved(P, degree(dbces11)) #>  TRUE is_preserved(P, closeness(dbces11)) #>  TRUE is_preserved(P, betweenness(dbces11)) #>  TRUE