hierarchy                package:sna                R Documentation

_C_o_m_p_u_t_e _G_r_a_p_h _H_i_e_r_a_r_c_h_y _S_c_o_r_e_s

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

     'hierarchy' takes a graph stack ('dat') and returns reciprocity or
     Krackhardt hierarchy scores for the graphs selected by 'g'.

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

     hierarchy(dat, g=1:stackcount(dat), measure=c("reciprocity", 
         "krackhardt"))

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

     dat: A graph or graph stack 

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

 measure: One of '"reciprocity"' or '"krackhardt"' 

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

     Hierarchy measures quantify the extent of asymmetry in a
     structure; the greater the extent of asymmetry, the more
     hierarchical the structure is said to be.  (This should not be
     confused with how _centralized_ the structure is, i.e., the extent
     to which centralities of vertex positions are highly
     concentrated.)  'hierarchy' provides two measures (selected by the
     'measure' argument) as follows:

        1.  'reciprocity': This setting returns the dyadic reciprocity
           for each input graph (see 'grecip')

        2.  'krackhardt': This setting returns the Krackhardt hierarchy
           score for each input graph.  The Krackhardt hierarchy is
           defined as the fraction of non-null dyads in the
           'reachability' graph which are asymmetric.  Thus, when no
           directed paths are reciprocated (e.g., in an in/outtree),
           Krackhardt hierarchy is equal to 1; when all such paths are
           reciprocated, by contrast (e.g., in a cycle or clique), the
           measure falls to 0. 

           Hierarchy 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.

     Note that hierarchy is inherently density-constrained: as
     densities climb above 0.5, the proportion of mutual dyads must (by
     the pigeonhole principle) increase rapidly, thereby reducing
     possibilities for asymmetry.  Thus, the interpretation of
     hierarchy scores should take density into account, particularly if
     density is artifactual (e.g., due to a particular dichotomization
     procedure).

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

     A vector of hierarchy 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.

_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. 

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

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

     'connectedness', 'efficiency', 'hierarchy', 'lubness', 'grecip',
     'mutuality', 'dyad.census'

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

     #Get hierarchy scores for graphs of varying densities
     hierarchy(rgraph(10,5,tprob=c(0.1,0.25,0.5,0.75,0.9)),
         measure="reciprocity")
     hierarchy(rgraph(10,5,tprob=c(0.1,0.25,0.5,0.75,0.9)),
         measure="krackhardt")

