Centralization is a method for creating a graph level centralization measure from the centrality scores of the vertices.
The vertex level centrality scores.
Real scalar. The graph level centrality score of
the most centralized graph with the same number of vertices as the graph
under study. This is only used if the
Logical scalar. Whether to normalize the graph level centrality score by dividing by the supplied theoretical maximum.
Centralization is a general method for calculating a graph-level centrality score based on node-level centrality measure. The formula for this is
C(G)=sum( max(c(w), w) - c(v),v),
where c(v) is the centrality of vertex v.
The graph-level centrality score can be normalized by dividing by the maximum theoretical score for a graph with the same number of vertices, using the same parameters, e.g. directedness, whether we consider loop edges, etc.
For degree, closeness and betweenness the most centralized structure is some version of the star graph, in-star, out-star or undirected star.
For eigenvector centrality the most centralized structure is the graph with a single edge (and potentially many isolates).
centralize implements general centralization formula to calculate
a graph-level score from vertex-level scores.
A real scalar, the centralization of the graph from which
scores were derived.
Freeman, L.C. (1979). Centrality in Social Networks I: Conceptual Clarification. Social Networks 1, 215–239.
Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press.
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# A BA graph is quite centralized g <- sample_pa(1000, m=4) centr_degree(g)$centralization centr_clo(g, mode="all")$centralization centr_eigen(g, directed=FALSE)$centralization # The most centralized graph according to eigenvector centrality g0 <- graph( c(2,1), n=10, dir=FALSE ) g1 <- make_star(10, mode="undirected") centr_eigen(g0)$centralization centr_eigen(g1)$centralization
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