code: def katz_centrality(
      G,
      alpha=None,
      beta=1.0,
      max_iter=100,
      tol=1.0e-6,
      nstart=None,
      normalized=True,
  ):
      """
      Compute the Katz centrality for the nodes of the graph G. This
      implementation is based on a relaxed version of Katz defined by Foster
      with a reduced computational complexity of O(n+m)
  
      On a directed graph, cuGraph computes the out-edge Katz centrality score.
      This is opposite of NetworkX which compute the in-edge Katz centrality
      score by default.  You can flip the NetworkX edges, using G.reverse,
      so that the results match cuGraph.
  
      References
      ----------
      Foster, K.C., Muth, S.Q., Potterat, J.J. et al.
      Computational & Mathematical Organization Theory (2001) 7: 275.
      https://doi.org/10.1023/A:1013470632383
  
      Katz, L. (1953). A new status index derived from sociometric analysis.
      Psychometrika, 18(1), 39-43.
  
      Parameters
      ----------
      G : cuGraph.Graph or networkx.Graph
          cuGraph graph descriptor with connectivity information. The graph can
          contain either directed or undirected edges.
  
      alpha : float, optional (default=None)
          Attenuation factor defaulted to None. If alpha is not specified then
          it is internally calculated as 1/(degree_max) where degree_max is the
          maximum out degree.
  
          NOTE:
              The maximum acceptable value of alpha for convergence
              alpha_max = 1/(lambda_max) where lambda_max is the largest
              eigenvalue of the graph.
              Since lambda_max is always lesser than or equal to degree_max for a
              graph, alpha_max will always be greater than or equal to
              (1/degree_max). Therefore, setting alpha to (1/degree_max) will
              guarantee that it will never exceed alpha_max thus in turn
              fulfilling the requirement for convergence.
  
      beta : float, optional (default=None)
          Weight scalar added to each vertex's new Katz Centrality score in every
          iteration. If beta is not specified then it is set as 1.0.
  
      max_iter : int, optional (default=100)
          The maximum number of iterations before an answer is returned. This can
          be used to limit the execution time and do an early exit before the
          solver reaches the convergence tolerance.
  
      tol : float, optional (default=1.0e-6)
          Set the tolerance the approximation, this parameter should be a small
          magnitude value.
          The lower the tolerance the better the approximation. If this value is
          0.0f, cuGraph will use the default value which is 1.0e-6.
          Setting too small a tolerance can lead to non-convergence due to
          numerical roundoff. Usually values between 1e-2 and 1e-6 are
          acceptable.
  
      nstart : cudf.Dataframe, optional (default=None)
          GPU Dataframe containing the initial guess for katz centrality.
  
          nstart['vertex'] : cudf.Series
              Contains the vertex identifiers
          nstart['values'] : cudf.Series
              Contains the katz centrality values of vertices
  
      normalized : not supported
          If True normalize the resulting katz centrality values
  
      Returns
      -------
      df : cudf.DataFrame or Dictionary if using NetworkX
          GPU data frame containing two cudf.Series of size V: the vertex
          identifiers and the corresponding katz centrality values.
  
          df['vertex'] : cudf.Series
              Contains the vertex identifiers
          df['katz_centrality'] : cudf.Series
              Contains the katz centrality of vertices
  
      Examples
      --------
      >>> from cugraph.datasets import karate
      >>> G = karate.get_graph(download=True)
      >>> kc = cugraph.katz_centrality(G)
  
      """
      G, isNx = ensure_cugraph_obj_for_nx(G, store_transposed=True)
  
      if G.store_transposed is False:
          warning_msg = (
              "Katz centrality expects the 'store_transposed' flag "
              "to be set to 'True' for optimal performance during "
              "the graph creation"
          )
          warnings.warn(warning_msg, UserWarning)
  
      if alpha is None:
          degree_max = G.degree()["degree"].max()
          alpha = 1 / (degree_max)
  
      if (alpha is not None) and (alpha <= 0.0):
          raise ValueError(
              f"'alpha' must be a positive float or None, "
              f"got: {alpha}"
          )
  
      elif (not isinstance(beta, float)) or (beta <= 0.0):
          raise ValueError(
              f"'beta' must be a positive float or None, "
              f"got: {beta}"
          )
      if (not isinstance(max_iter, int)) or (max_iter <= 0):
          raise ValueError(
              f"'max_iter' must be a positive integer"
              f", got: {max_iter}"
          )
      if (not isinstance(tol, float)) or (tol <= 0.0):
          raise ValueError(
              f"'tol' must be a positive float, got: {tol}"
          )
  
      if nstart is not None:
          if G.renumbered is True:
              if len(G.renumber_map.implementation.col_names) > 1:
                  cols = nstart.columns[:-1].to_list()
              else:
                  cols = "vertex"
              nstart = G.add_internal_vertex_id(nstart, "vertex", cols)
              nstart = nstart[nstart.columns[0]]
  
      vertices, values = pylibcugraph_katz(
          resource_handle=ResourceHandle(),
          graph=G._plc_graph,
          betas=nstart,
          alpha=alpha,
          beta=beta,
          epsilon=tol,
          max_iterations=max_iter,
          do_expensive_check=False,
      )
  
      vertices = cudf.Series(vertices)
      values = cudf.Series(values)
  
      df = cudf.DataFrame()
      df["vertex"] = vertices
      df["katz_centrality"] = values
  
      if G.renumbered:
          df = G.unrenumber(df, "vertex")
  
      if isNx is True:
          dict = df_score_to_dictionary(df, "katz_centrality")
          return dict
      else:
          return df
  
imports: from pylibcugraph import katz_centrality, ResourceHandle
  from cugraph.utilities import ensure_cugraph_obj_for_nx, df_score_to_dictionary
  import cudf
  import warnings
label_desc: Use image node_id 1 for calling a global function with example usage: katz_centrality(G, alpha, beta, max_iter, tol, nstart, normalized) and returns: dict, df