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 ctb@andrew.cmu.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)

