infocent                 package:sna                 R Documentation

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

     'infocent' takes a graph stack ('dat') and returns the information
     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:

     infocent(dat, g=1, nodes=c(1:dim(dat)[2]), gmode="digraph", 
         diag=FALSE, cmode="weak", tmaxdev=FALSE, rescale=FALSE,tol=1e-20)

_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. 

   cmode: The rule to be used by 'symmetrize' when symmetrizing
          dichotomous data; must be one of '"weak"' (for an 'OR' rule),
          '"strong"' for an 'AND' rule), '"upper"' (for a 'max' rule),
          or '"lower"' (for a 'min' rule).  Set to '"weak"' by default,
          this parameter obviously has no effect on symmetric data. 

 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. 

     tol: Tolerance for near-singularities during matrix inversion (see
          'solve') 

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

     Actor information centrality is a hybrid measure which relates to
     both path-length indices (e.g., closeness, graph centrality) and
     to walk-based eigenmeasures (e.g., eigenvector centrality,
     Bonacich power).  In particular, the information centrality of a
     given actor can be understood to be the harmonic average of the
     ``bandwidth'' for all paths originating with said individual
     (where the bandwidth is taken to be inversely related to path
     length).  Formally, the index is constructed as follows.  First,
     we take G to be an undirected (but possibly valued) graph -
     symmetrizing if necessary - with (possibly valued) adjacency
     matrix A.  From this, we remove all isolates (whose information
     centralities are zero in any event) and proceed to create the
     weighted connection matrix


                               C = B^-1


     where B is a pseudo-adjacency matrix formed by replacing the
     diagonal of 1-A with one plus each actor's degree.  Given the
     above, let T be the trace of C with sum S_T, and let S_R be an
     arbitrary row sum (all rows of C have the same sum).  The
     information centrality scores are then equal to


                  C_I = ( T + (S_T-2S_R)/|V(G)| )^-1


     (recalling that the scores for any omitted vertices are 0).

     In general, actors with higher information centrality are
     predicted to have greater control over the flow of information
     within a network; highly information-central individuals tend to
     have a large number of short paths to many others within the
     social structure.  Because the raw centrality values can be
     difficult to interpret directly, rescaled values are sometimes
     preferred (see the 'rescale' option).  Though the use of path
     weights suggest information centrality as a possible replacement
     for closeness, the problem of inverting the B matrix poses
     problems of its own; as with all such measures, caution is advised
     on disconnected or degenerate structures.

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

     A vector containing the centrality scores

_N_o_t_e:

     The theoretical maximum deviation used here is not obtained with
     the star network; rather, the maximum occurs for an empty graph
     with one complete dyad, which is the model used here.

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

     David Barron david.barron@jesus.ox.ac.uk

     Carter T. Butts buttsc@uci.edu

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

     Stephenson, K., and Zelen, M. (1989).  ``Rethinking Centrality:
     Methods and Applications.''  _Social Networks_, 11, 1-37.

     Wasserman, S., and Faust, K.  (1994).  ``Social Network Analysis:
     Methods and Applications.''  Cambridge: Cambridge University
     Press.

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

     'evcent', 'bonpow', 'closeness', 'graphcent', 'centralization'

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

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

