Centralization of a graphSource:
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 centralization measure of the most centralized graph with the same number of vertices as the graph under study. This is only used if the
normalizedargument is set to
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 $$C(G)=\sum_v (\max_w c_w - c_v),$$ where \(c_v\) is the centrality of vertex \(v\).
The graph-level centralization measure 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.
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.
# A BA graph is quite centralized g <- sample_pa(1000, m = 4) centr_degree(g)$centralization #>  0.1543185 centr_clo(g, mode = "all")$centralization #>  0.4168858 centr_eigen(g, directed = FALSE)$centralization #>  0.9423069 # Calculate centralization from pre-computed scores deg <- degree(g) tmax <- centr_degree_tmax(g, loops = FALSE) centralize(deg, tmax) #>  0.1544731 # The most centralized graph according to eigenvector centrality g0 <- make_graph(c(2, 1), n = 10, dir = FALSE) g1 <- make_star(10, mode = "undirected") centr_eigen(g0)$centralization #>  1 centr_eigen(g1)$centralization #>  0.75