brokerage                package:sna                R Documentation

_P_e_r_f_o_r_m _a _G_o_u_l_d-_F_e_r_n_a_n_d_e_z _B_r_o_k_e_r_a_g_e _A_n_a_l_y_s_i_s

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

     Performs the brokerage analysis of Gould and Fernandez on one or
     more input graphs, given a class membership vector.

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

     brokerage(g, cl)

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

       g: one or more input graphs. 

      cl: a vector of class memberships. 

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

     Gould and Fernandez (following Marsden and others) describe
     _brokerage_ as the role played by a social actor who mediates
     contact between two alters.  More formally, vertex v is a broker
     for distinct vertices a and b iff a -> v -> b and a -!> b.  Where
     actors belong to a priori distinct groups, group membership may be
     used to segment brokerage roles into particular types.  Let A -> B
     -> C denote the two-path associated with a brokerage structure,
     such that some vertex from group B brokers the connection from
     some vertex from group A to a vertex in group C.  The types of
     brokerage roles defined by Gould and Fernandez (and their
     accompanying two-path structures) are then defined in terms of
     group membership as follows:

          \item[w_I]{Coordinator role; the broker mediates contact
          between two individuals from his or her own group.  Two-path
          structure: A -> A -> A}   \item[w_O]{Itinerant broker role;
          the broker mediates contact between two individuals from a
          single group to which he or she does not belong.  Two-path
          structure: A -> B -> A}   \item[b_{IO}]{Gatekeeper role; the
          broker mediates an incoming contact from an out-group member
          to an in-group member.  Two-path structure: A -> B -> B}  
          \item[b_{OI}]{Representative role; the broker mediates an
          outgoing contact from an in-group member to an out-group
          member.  Two-path structure: A -> A -> B}  
          \item[b_O]{Liaison role; the broker mediates contact between
          two individuals from different groups, neither of which is
          the group to which he or she belongs.  Two-path structure: A
          -> B -> C}   \item[t]{Total (cumulative) brokerage role
          occupancy.  (Any of the above two-paths.)}  

     The _brokerage score_ for a given vertex with respect to a given
     role is the number of ordered pairs having the appropriate group
     membership(s) brokered by said vertex.  'brokerage' computes the
     brokerage scores for each vertex, given an input graph and vector
     of class memberships.  Aggregate scores are also computed at the
     graph level, which correspond to the total frequency of each role
     type within the network structure.  Expectations and variances of
     the brokerage scores conditional on size and density are computed,
     along with approximate $z$-tests for incidence of brokerage. 
     (Note that the accuracy of the normality assumption is not known
     in the general case; see Gould and Fernandez (1989) for details. 
     Simulation-based tests may be desirable as an alternative.)

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

     An object of class 'brokerage', containing the following elements: 

raw.nli : The matrix of observed brokerage scores, by vertex

exp.nli : The matrix of expected brokerage scores, by vertex

 sd.nli : The matrix of predicted brokerage score standard deviations,
          by vertex

  z.nli : The matrix of standardized brokerage scores, by vertex

raw.gli : The vector of observed aggregate brokerage scores

exp.gli : The vector of expected aggregate brokerage scores

 sd.gli : The vector of predicted aggregate brokerage score standard
          deviations

  z.gli : The vector of standardized aggregate brokerage scores

exp.grp : The matrix of expected brokerage scores, by group

 sd.grp : The matrix of predicted brokerage score standard deviations,
          by group

     cl : The vector of class memberships

   clid : The original class names

      n : The input class sizes

      N : The order of the input network

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

     Carter T. Butts buttsc@uci.edu

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

     Gould, R.V. and Fernandez, R.M.  1989.  "Structures of Mediation:
     A Formal Approach to Brokerage in Transaction Networks." 
     _Sociological Methodology,_ 19: 89-126.

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

     'triad.census', 'gtrans'

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

     #Draw a random network with 3 groups
     g<-rgraph(15)
     cl<-rep(1:3,5)

     #Compute a brokerage object
     b<-brokerage(g,cl)
     summary(b)

