efficiency                package:sna                R Documentation

_C_o_m_p_u_t_e _G_r_a_p_h _E_f_f_i_c_i_e_n_c_y _S_c_o_r_e_s

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

     'efficiency' takes one or more graphs ('dat') and returns the
     Krackhardt efficiency scores for the graphs selected by 'g'.

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

     efficiency(dat, g=NULL, diag=FALSE)

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

     dat: one or more graphs. 

       g: index values for the graphs to be utilized; by default, all
          graphs are selected. 

    diag: 'TRUE' if the diagonal contains valid data; by default,
          'diag==FALSE'. 

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

     Let G= G_1 U ... U G_n be a digraph with weak components
     G_1,G_2,...,G_n.  For convenience, we denote the cardinalities of
     these components' vertex sets by |V(G)|=N and |V(G_i)|=N_i, for i
     in 1,...,n.  Then the Krackhardt efficiency of G is given by


 1 - ( |E(G)| - Sum(N_i-1,i=1,..,n) )/( Sum(N_i(N_i-1) - (N_i-1),i=1,..,n) )


     which can be interpreted as 1 minus the proportion of possible
     ``extra'' edges (above those needed to weakly connect the existing
     components) actually present in the graph.  A graph which an
     efficiency of 1 has precisely as many edges as are needed to
     connect its components; as additional edges are added, efficiency
     gradually falls towards 0.

     Efficiency is one of four measures ('connectedness', 'efficiency',
     'hierarchy', and 'lubness') suggested by Krackhardt for
     summarizing hierarchical structures.  Each corresponds to one of
     four axioms which are necessary and sufficient for the structure
     in question to be an outtree; thus, the measures will be equal to
     1 for a given graph iff that graph is an outtree.  Deviations from
     unity can be interpreted in terms of failure to satisfy one or
     more of the outtree conditions, information which may be useful in
     classifying its structural properties.

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

     A vector of efficiency scores

_N_o_t_e:

     The four Krackhardt indices are, in general, nondegenerate for a
     relatively narrow band of size/density combinations (efficiency
     being the sole exception).  This is primarily due to their
     dependence on the reachability graph, which tends to become
     complete rapidly as size/density increase.  See Krackhardt (1994)
     for a useful simulation study. 

     The violation normalization used before version 0.51 was N(N-1) -
     Sum(N_i-1,i=1,..,n), based on a somewhat different interpretation
     of the definition in Krackhardt (1994).  The former version gave
     results which more closely matched those of the cited simulation
     study, but was less consistent with the textual definition.

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

     Carter T. Butts buttsc@uci.edu

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

     Krackhardt, David.  (1994).  ``Graph Theoretical Dimensions of
     Informal Organizations.'' In K. M. Carley and M. J. Prietula
     (Eds.), _Computational Organization Theory_, 89-111. Hillsdale,
     NJ: Lawrence Erlbaum and Associates.

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

     'connectedness', 'efficiency', 'hierarchy', 'lubness', 'gden'

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

     #Get efficiency scores for graphs of varying densities
     efficiency(rgraph(10,5,tprob=c(0.1,0.25,0.5,0.75,0.9)))

