centralization              package:sna              R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

     `Centralization' returns the centralization GLI (graph-level
     index) for a given graph in `dat', given a (node) centrality
     measure `FUN'.  `Centralization' follows Freeman's (1979)
     generalized definition of network centralization, and can be used
     with any properly defined centrality measure.  This measure must
     be implemented separately; see the references below for examples.

_U_s_a_g_e:

     centralization(dat, FUN, g=1, mode="digraph", diag=FALSE, 
         normalize=TRUE, ...)

_A_r_g_u_m_e_n_t_s:

     dat: Data array to be analyzed.  By assumption, the first
          dimension of the array indexes the graph, with the next two
          indexing the actors.  Provided that `FUN' is well-behaved,
          this can be an n x n matrix if only one graph is involved. 

     FUN: Function to return nodal centrality scores.

       g: Integer indicating the index of the graph for which
          centralization should be computed.  By default, `g'=1. 

    mode: String indicating the type of graph being evaluated. 
          "digraph" indicates that edges should be interpreted as
          directed; "graph" indicates that edges are undirected. 
          `mode' is set to "digraph" by default. 

    diag: Boolean indicating whether or not the diagonal should be
          treated as valid data.  Set this true if and only if the data
          can contain loops.  `diag' is `FALSE' by default. 

normalize: Boolean indicating whether or not the centralization score
          should be normalized to the theoretical maximum.  (Note that
          this function relies on `FUN' to return this value when
          called with `tmaxdev==TRUE'.)  By default, `tmaxdev==TRUE'. 

     ...: Additional arguments to `FUN'. 

_D_e_t_a_i_l_s:

     The centralization of a graph G for centrality measure C(v) is
     defined (as per Freeman (1979)) to be:


             C^*(G) = sum( |max(C(v))-C(i)|, i in V(G) )


     Or, equivalently, the absolute deviation from the maximum of C on
     G.  Generally, this value is normalized by the theoretical maximum
     centralization score, conditional on |V(G)|.  (Here, this
     functionality is activated by `normalize'.)  `Centralization'
     depends on the function specified by `FUN' to return the vector of
     nodal centralities when called with `dat' and `g', and to return
     the theoretical maximum value when called with the above and
     `tmaxdev==TRUE'.  For an example of such a centrality routine, see
     `degree'.

_V_a_l_u_e:

     The centralization of the specified graph.

_N_o_t_e:

     See `cugtest' for null hypothesis tests involving centralization
     scores.

_A_u_t_h_o_r(_s):

     Carter T. Butts ctb@andrew.cmu.edu

_R_e_f_e_r_e_n_c_e_s:

     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: Cambridge University Press.

_S_e_e _A_l_s_o:

     `cugtest'

_E_x_a_m_p_l_e_s:

     #Generate some random graphs
     dat<-rgraph(5,10)
     #How centralized is the third one on indegree?
     centralization(dat,g=3,degree,cmode="indegree")
     #How about on total (Freeman) degree?
     centralization(dat,g=3,degree)

