lubness                 package:sna                 R Documentation

_C_o_m_p_u_t_e _G_r_a_p_h _L_U_B_n_e_s_s _S_c_o_r_e_s

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

     'lubness' takes a graph stack ('dat') and returns the Krackhardt
     LUBness scores for the graphs selected by 'g'.

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

     lubness(dat, g=1:stackcount(dat))

_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 

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

     In the context of a directed graph G, two actors i and j may be
     said to have an _upper bound_ iff there exists some actor k such
     that directed ki and kj paths are belong to G.  An upper bound ell
     is known as a _least upper bound_ for i and j iff it belongs to at
     least one ki and kj path (respectively) for all i,j upper bounds
     k; let L(i,j) be an indicator which returns 1 iff such an $\ell$
     exists, otherwise returning 0.  Now, let G_1,G_2,...,G_n represent
     the weak components of G.  For convenience, we denote the
     cardinalities of these graphs' vertex sets by |V(G)|=N and
     |V(G_i)|=N_i, for i in 1,...,n.  Given this, the Krackhardt
     LUBness of G is given by


 1-Sum(Sum(1-L(v_j,v_k),v_j,v_k in V(G_i)),i=1,...,n)/Sum((N_i-1)(N_i-2)/2,i=1,...,n)


     Where all vertex pairs possess a least upper bound, Krackhardt's
     LUBness is equal to 1; in general, it approaches 0 as this
     condition is broached.  (This convergence is problematic in
     certain cases due to the requirement that we sum violations across
     components; where a graph contains no components of size three or
     greater, Krackhardt's LUBness is not well-defined.  'lubness'
     returns a 'NaN' in these cases.)  

     LUBness 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 LUBness 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.

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

     'connectedness', 'efficiency', 'hierarchy', 'lubness',
     'reachability'

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

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

