dev/R_igraph/centralization.R
In kbodwin/longnet: What the Package Does (One Line, Title Case)
## -----------------------------------------------------------------------
##
## IGraph R package
## Copyright (C) 2015 Gabor Csardi <csardi.gabor@gmail.com>
## 334 Harvard street, Cambridge, MA 02139 USA
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, write to the Free Software
## Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
## 02110-1301 USA
##
## -----------------------------------------------------------------------
#' @include auto.R
NULL
#' Centralization of a graph
#'
#' Centralization is a method for creating a graph level centralization
#' measure from the centrality scores of the vertices.
#'
#' Centralization is a general method for calculating a graph-level
#' centrality score based on node-level centrality measure. The formula for
#' this is
#' \deqn{C(G)=\sum_v (\max_w c_w - c_v),}{ C(G)=sum( max(c(w), w) - c(v),v),}
#' where \eqn{c_v}{c(v)} is the centrality of vertex \eqn{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).
#'
#' \code{centralize} implements general centralization formula to calculate
#' a graph-level score from vertex-level scores.
#'
#' @param scores The vertex level centrality scores.
#' @param theoretical.max 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 \code{normalized} argument is set
#' to \code{TRUE}.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the supplied theoretical maximum.
#' @return A real scalar, the centralization of the graph from which
#' \code{scores} were derived.
#'
#' @aliases centralization centralize.scores
#' @family centralization related
#'
#' @export
#' @references Freeman, L.C. (1979). Centrality in Social Networks I:
#' Conceptual Clarification. \emph{Social Networks} 1, 215--239.
#'
#' Wasserman, S., and Faust, K. (1994). \emph{Social Network Analysis:
#' Methods and Applications.} Cambridge University Press.
#'
#' @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)$centralization
#'
#' # The most centralized graph according to eigenvector centrality
#' g0 <- graph( c(2,1), n=10, dir=FALSE )
#' g1 <- make_star(10, mode="undirected")
#' centr_eigen(g0)$centralization
#' centr_eigen(g1)$centralization
centralize <- centralize
#' Centralize a graph according to the degrees of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param mode This is the same as the \code{mode} argument of
#' \code{degree}.
#' @param loops Logical scalar, whether to consider loops edges when
#' calculating the degree.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.degree
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_degree <- centr_degree
#' Theoretical maximum for degree centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes}, \code{mode} and \code{loops} are all given.
#' @param nodes The number of vertices. This is ignored if the graph is given.
#' @param mode This is the same as the \code{mode} argument of
#' \code{degree}.
#' @param loops Logical scalar, whether to consider loops edges when
#' calculating the degree.
#' @return Real scalar, the theoratical maximum (unnormalized) graph degree
#' centrality score for graphs with given order and other parameters.
#'
#' @aliases centralization.degree.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g, normalized = FALSE)$centralization %>%
#' `/`(centr_degree_tmax(g))
#' centr_degree(g, normalized = TRUE)$centralization
centr_degree_tmax <- centr_degree_tmax
#' Centralize a graph according to the betweenness of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param directed logical scalar, whether to use directed shortest paths for
#' calculating betweenness.
#' @param nobigint Logical scalar, whether to use big integers for the
#' betweenness calculation. This argument is passed to the
#' \code{\link{betweenness}} function.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.betweenness
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_betw <- centr_betw
#' Theoretical maximum for betweenness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param directed logical scalar, whether to use directed shortest paths
#' for calculating betweenness.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' betweenness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.betweenness.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_betw(g, normalized = FALSE)$centralization %>%
#' `/`(centr_betw_tmax(g))
#' centr_betw(g, normalized = TRUE)$centralization
centr_betw_tmax <- centr_betw_tmax
#' Centralize a graph according to the closeness of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param mode This is the same as the \code{mode} argument of
#' \code{closeness}.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.closeness
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_clo <- centr_clo
#' Theoretical maximum for closeness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param mode This is the same as the \code{mode} argument of
#' \code{closeness}.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' closeness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.closeness.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_clo(g, normalized = FALSE)$centralization %>%
#' `/`(centr_clo_tmax(g))
#' centr_clo(g, normalized = TRUE)$centralization
centr_clo_tmax <- centr_clo_tmax
#' Centralize a graph according to the eigenvector centrality of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param directed logical scalar, whether to use directed shortest paths for
#' calculating eigenvector centrality.
#' @param scale Whether to rescale the eigenvector centrality scores, such that
#' the maximum score is one.
#' @param options This is passed to \code{\link{eigen_centrality}}, the options
#' for the ARPACK eigensolver.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{vector}{The node-level centrality scores.}
#' \item{value}{The corresponding eigenvalue.}
#' \item{options}{ARPACK options, see the return value of
#' \code{\link{eigen_centrality}} for details.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The same as above, the theoretical maximum
#' centralization score for a graph with the same number of vertices.}
#'
#' @aliases centralization.evcent
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
#'
#' # The most centralized graph according to eigenvector centrality
#' g0 <- make_graph(c(2,1), n = 10, dir = FALSE)
#' g1 <- make_star(10, mode = "undirected")
#' centr_eigen(g0)$centralization
#' centr_eigen(g1)$centralization
centr_eigen <- centr_eigen
#' Theoretical maximum for betweenness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param directed logical scalar, whether to use directed shortest paths
#' for calculating betweenness.
#' @param scale Whether to rescale the eigenvector centrality scores,
#' such that the maximum score is one.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' betweenness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.evcent.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_eigen(g, normalized = FALSE)$centralization %>%
#' `/`(centr_eigen_tmax(g))
#' centr_eigen(g, normalized = TRUE)$centralization
centr_eigen_tmax <- centr_eigen_tmax
kbodwin/longnet documentation built on Nov. 4, 2019, 3:33 p.m.
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## -----------------------------------------------------------------------
##
## IGraph R package
## Copyright (C) 2015 Gabor Csardi <csardi.gabor@gmail.com>
## 334 Harvard street, Cambridge, MA 02139 USA
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, write to the Free Software
## Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
## 02110-1301 USA
##
## -----------------------------------------------------------------------
#' @include auto.R
NULL
#' Centralization of a graph
#'
#' Centralization is a method for creating a graph level centralization
#' measure from the centrality scores of the vertices.
#'
#' Centralization is a general method for calculating a graph-level
#' centrality score based on node-level centrality measure. The formula for
#' this is
#' \deqn{C(G)=\sum_v (\max_w c_w - c_v),}{ C(G)=sum( max(c(w), w) - c(v),v),}
#' where \eqn{c_v}{c(v)} is the centrality of vertex \eqn{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).
#'
#' \code{centralize} implements general centralization formula to calculate
#' a graph-level score from vertex-level scores.
#'
#' @param scores The vertex level centrality scores.
#' @param theoretical.max 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 \code{normalized} argument is set
#' to \code{TRUE}.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the supplied theoretical maximum.
#' @return A real scalar, the centralization of the graph from which
#' \code{scores} were derived.
#'
#' @aliases centralization centralize.scores
#' @family centralization related
#'
#' @export
#' @references Freeman, L.C. (1979). Centrality in Social Networks I:
#' Conceptual Clarification. \emph{Social Networks} 1, 215--239.
#'
#' Wasserman, S., and Faust, K. (1994). \emph{Social Network Analysis:
#' Methods and Applications.} Cambridge University Press.
#'
#' @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
#'
#' # 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
centralize <- centralize
#' Centralize a graph according to the degrees of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param mode This is the same as the \code{mode} argument of
#' \code{degree}.
#' @param loops Logical scalar, whether to consider loops edges when
#' calculating the degree.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.degree
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_degree <- centr_degree
#' Theoretical maximum for degree centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes}, \code{mode} and \code{loops} are all given.
#' @param nodes The number of vertices. This is ignored if the graph is given.
#' @param mode This is the same as the \code{mode} argument of
#' \code{degree}.
#' @param loops Logical scalar, whether to consider loops edges when
#' calculating the degree.
#' @return Real scalar, the theoratical maximum (unnormalized) graph degree
#' centrality score for graphs with given order and other parameters.
#'
#' @aliases centralization.degree.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g, normalized = FALSE)$centralization %>%
#' `/`(centr_degree_tmax(g))
#' centr_degree(g, normalized = TRUE)$centralization
centr_degree_tmax <- centr_degree_tmax
#' Centralize a graph according to the betweenness of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param directed logical scalar, whether to use directed shortest paths for
#' calculating betweenness.
#' @param nobigint Logical scalar, whether to use big integers for the
#' betweenness calculation. This argument is passed to the
#' \code{\link{betweenness}} function.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.betweenness
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_betw <- centr_betw
#' Theoretical maximum for betweenness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param directed logical scalar, whether to use directed shortest paths
#' for calculating betweenness.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' betweenness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.betweenness.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_betw(g, normalized = FALSE)$centralization %>%
#' `/`(centr_betw_tmax(g))
#' centr_betw(g, normalized = TRUE)$centralization
centr_betw_tmax <- centr_betw_tmax
#' Centralize a graph according to the closeness of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param mode This is the same as the \code{mode} argument of
#' \code{closeness}.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{res}{The node-level centrality scores.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The maximum theoretical graph level
#' centralization score for a graph with the given number of vertices,
#' using the same parameters. If the \code{normalized} argument was
#' \code{TRUE}, then the result was divided by this number.}
#'
#' @aliases centralization.closeness
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
centr_clo <- centr_clo
#' Theoretical maximum for closeness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param mode This is the same as the \code{mode} argument of
#' \code{closeness}.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' closeness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.closeness.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_clo(g, normalized = FALSE)$centralization %>%
#' `/`(centr_clo_tmax(g))
#' centr_clo(g, normalized = TRUE)$centralization
centr_clo_tmax <- centr_clo_tmax
#' Centralize a graph according to the eigenvector centrality of vertices
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph.
#' @param directed logical scalar, whether to use directed shortest paths for
#' calculating eigenvector centrality.
#' @param scale Whether to rescale the eigenvector centrality scores, such that
#' the maximum score is one.
#' @param options This is passed to \code{\link{eigen_centrality}}, the options
#' for the ARPACK eigensolver.
#' @param normalized Logical scalar. Whether to normalize the graph level
#' centrality score by dividing by the theoretical maximum.
#' @return A named list with the following components:
#' \item{vector}{The node-level centrality scores.}
#' \item{value}{The corresponding eigenvalue.}
#' \item{options}{ARPACK options, see the return value of
#' \code{\link{eigen_centrality}} for details.}
#' \item{centralization}{The graph level centrality index.}
#' \item{theoretical_max}{The same as above, the theoretical maximum
#' centralization score for a graph with the same number of vertices.}
#'
#' @aliases centralization.evcent
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_degree(g)$centralization
#' centr_clo(g, mode = "all")$centralization
#' centr_betw(g, directed = FALSE)$centralization
#' centr_eigen(g, directed = FALSE)$centralization
#'
#' # 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)$centralization
#' centr_eigen(g1)$centralization
centr_eigen <- centr_eigen
#' Theoretical maximum for betweenness centralization
#'
#' See \code{\link{centralize}} for a summary of graph centralization.
#'
#' @param graph The input graph. It can also be \code{NULL}, if
#' \code{nodes} is given.
#' @param nodes The number of vertices. This is ignored if the graph is
#' given.
#' @param directed logical scalar, whether to use directed shortest paths
#' for calculating betweenness.
#' @param scale Whether to rescale the eigenvector centrality scores,
#' such that the maximum score is one.
#' @return Real scalar, the theoratical maximum (unnormalized) graph
#' betweenness centrality score for graphs with given order and other
#' parameters.
#'
#' @aliases centralization.evcent.tmax
#' @family centralization related
#'
#' @export
#'
#' @examples
#' # A BA graph is quite centralized
#' g <- sample_pa(1000, m = 4)
#' centr_eigen(g, normalized = FALSE)$centralization %>%
#' `/`(centr_eigen_tmax(g))
#' centr_eigen(g, normalized = TRUE)$centralization
centr_eigen_tmax <- centr_eigen_tmax
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