evcent                  package:sna                  R Documentation

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

     `evcent' takes a graph stack (`dat') and returns the eigenvector
     centralities of positions within one graph (indicated by `nodes'
     and `g', respectively).  This function is compatible with
     `centralization', and will return the theoretical maximum absolute
     deviation (from maximum) conditional on size (which is used by
     `centralization' to normalize the observed centralization score).

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

     evcent(dat, g=1, nodes=c(1:dim(dat)[2]), gmode="digraph", diag=FALSE,
         tmaxdev=FALSE, rescale=FALSE)

_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. Alternately, this can be an n x n matrix
          (if only one graph is involved). 

       g: Integer indicating the index of the graph for which
          centralities are to be calculated.  By default, `g==1'. 

   nodes: List indicating which nodes are to be included in the
          calculation.  By default, all nodes are included. 

   gmode: String indicating the type of graph being evaluated. 
          "digraph" indicates that edges should be interpreted as
          directed; "graph" indicates that edges are undirected.  This
          is currently ignored. 

    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. 

 tmaxdev: Boolean indicating whether or not the theoretical maximum
          absolute deviation from the maximum nodal centrality should
          be returned.  By default, `tmaxdev==FALSE'. 

 rescale: If true, centrality scores are rescaled such that they sum to
          1. 

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

     Eigenvector centrality scores correspond to the values of the
     first eigenvector of the graph adjacency matrix; these scores may,
     in turn, be interpreted as arising from a reciprocal process in
     which the centrality of each actor is proportional to the sum of
     the centralities of those actors to whom he or she is connected. 
     In general, vertices with high eigenvector centralities are those
     which are connected to many other vertices which are, in turn,
     connected to many others (and so on).  (The perceptive may realize
     that this implies that the largest values will be obtained by
     individuals in large cliques (or high-density substructures). 
     This is also intelligible from an algebraic point of view, with
     the first eigenvector being closely related to the best rank-1
     approximation of the adjacency matrix (a relationship which is
     easy to see in the special case of a diagonalizable symmetric real
     matrix via the SLS^-1 decomposition).) 

     The simple eigenvector centrality is generalized by the Bonacich
     power centrality measure; see `bonpow' for more details.

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

     A vector containing the centrality scores

_W_A_R_N_I_N_G:

     `evcent' will not symmetrize your data before extracting
     eigenvectors; don't send this routine asymmetric matrices unless
     you really mean to do so.

_N_o_t_e:

     The theoretical maximum deviation used here is not obtained with
     the star network, in general.  For positive exponents, at least,
     the maximum occurs for an empty graph with one complete dyad. 
     UCINET V seems not to adjust for this fact, which can cause some
     oddities in their 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:

     Bonacich, P.  (1987).  ``Power and Centrality: A Family of
     Measures.'' American Journal of Sociology, 92, 1170-1182.

     Katz, L.  (1953).  ``A New Status Index Derived from Sociometric
     Analysis.''  Psychometrika, 18, 39-43.

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

     `centralization', `bonpow'

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

     #Generate some test data
     dat<-rgraph(10,mode="graph")
     #Compute eigenvector centrality scores
     evcent(dat)

