igraph (version 1.3.5)

centralize: Centralization of a graph

Description

Centralization is a method for creating a graph level centralization measure from the centrality scores of the vertices.

Usage

centralize(scores, theoretical.max = 0, normalized = TRUE)

Value

A real scalar, the centralization of the graph from which

scores were derived.

Arguments

scores

The vertex level centrality scores.

theoretical.max

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 normalized argument is set to TRUE.

normalized

Logical scalar. Whether to normalize the graph level centrality score by dividing by the supplied theoretical maximum.

Details

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.

References

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.

See Also

Other centralization related: centr_betw_tmax(), centr_betw(), centr_clo_tmax(), centr_clo(), centr_degree_tmax(), centr_degree(), centr_eigen_tmax(), centr_eigen()

Examples

# 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

# Calculate centralization from pre-computed scores
deg <- degree(g)
tmax <- centr_degree_tmax(g, loops=FALSE)
centralize(deg, tmax)

# 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