alpha.centrality           package:igraph           R Documentation

_F_i_n_d _B_o_n_a_c_i_c_h _a_l_p_h_a _c_e_n_t_r_a_l_i_t_y _s_c_o_r_e_s _o_f _n_e_t_w_o_r_k _p_o_s_i_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     'alpha.centrality' calculates the alpha centrality of some (or
     all) vertices in a graph.

_U_s_a_g_e:

     alpha.centrality(graph, nodes=V(graph), alpha=1, loops=FALSE,
                      exo=1, weights=NULL, tol=1e-7)

_A_r_g_u_m_e_n_t_s:

   graph: The input graph, can be directed or undirected

   nodes: Vertex sequence, the vertices for which the alpha centrality
          values are returned. (For technical reasons they will be
          calculated for all vertices first, anyway.)

   alpha: Parameter specifying the relative importance of endogenous
          versus exogenous factors in the determination of centrality.
          See details below.

   loops: Whether to eliminate loop edges from the graph before the
          calculation.

     exo: The exogenous factors, in most cases this is either a
          constant - the same factor for every node, or a vector giving
          the factor for every vertex. Note that long vectors will be
          truncated and short vectors will be replicated.

 weights: Optional positive weight vector for calculating weighted
          closeness. If the graph has a 'weight' edge attribute, then
          this is used by default.

     tol: Tolerance for near-singularities during matrix inversion, see
          'solve'.

_D_e_t_a_i_l_s:

     The alpha centrality measure can be considered as a generalization
     of eigenvector centerality to directed graphs. It was proposed by
     Bonacich in 2001 (see reference below).

     The alpha centrality of the vertices in a graph is defined as the
     solution of the following matrix equation:

                           x=alpha t(A)x+e,

     where A is the (not neccessarily symmetric) adjacency matrix of
     the graph, e is the vector of exogenous sources of status of the
     vertices and alpha is the relative importance of the endogenous
     versus exogenous factors.

_V_a_l_u_e:

     A numeric vector contaning the centrality scores for the selected
     vertices.

_W_a_r_n_i_n_g:

     Singular adjacency matrices cause problems for this algorithm, the
     routine may fail is certain cases.

_A_u_t_h_o_r(_s):

     Gabor Csardi csardi@rmki.kfki.hu.

_R_e_f_e_r_e_n_c_e_s:

     Bonacich, P. and Paulette, L. (2001). ``Eigenvector-like measures
     of centrality for asymmetric relations'' _Social Networks_, 23,
     191-201.

_S_e_e _A_l_s_o:

     'evcent' and 'bonpow'

_E_x_a_m_p_l_e_s:

     # The examples from Bonacich's paper
     g.1 <- graph( c(1,3,2,3,3,4,4,5)-1 )
     g.2 <- graph( c(2,1,3,1,4,1,5,1)-1 )
     g.3 <- graph( c(1,2,2,3,3,4,4,1,5,1)-1 )
     alpha.centrality(g.1)
     alpha.centrality(g.2)
     alpha.centrality(g.3,alpha=0.5)

