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