code: def katz_centrality(
      G,
      alpha=0.1,
      beta=1.0,
      max_iter=1000,
      tol=1.0e-6,
      nstart=None,
      normalized=True,
      weight=None,
  ):
      r"""Compute the Katz centrality for the nodes of the graph G.
  
      Katz centrality computes the centrality for a node based on the centrality
      of its neighbors. It is a generalization of the eigenvector centrality. The
      Katz centrality for node $i$ is
  
      .. math::
  
          x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
  
      where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.
  
      The parameter $\beta$ controls the initial centrality and
  
      .. math::
  
          \alpha < \frac{1}{\lambda_{\max}}.
  
      Katz centrality computes the relative influence of a node within a
      network by measuring the number of the immediate neighbors (first
      degree nodes) and also all other nodes in the network that connect
      to the node under consideration through these immediate neighbors.
  
      Extra weight can be provided to immediate neighbors through the
      parameter $\beta$.  Connections made with distant neighbors
      are, however, penalized by an attenuation factor $\alpha$ which
      should be strictly less than the inverse largest eigenvalue of the
      adjacency matrix in order for the Katz centrality to be computed
      correctly. More information is provided in [1]_.
  
      Parameters
      ----------
      G : graph
        A NetworkX graph.
  
      alpha : float, optional (default=0.1)
        Attenuation factor
  
      beta : scalar or dictionary, optional (default=1.0)
        Weight attributed to the immediate neighborhood. If not a scalar, the
        dictionary must have a value for every node.
  
      max_iter : integer, optional (default=1000)
        Maximum number of iterations in power method.
  
      tol : float, optional (default=1.0e-6)
        Error tolerance used to check convergence in power method iteration.
  
      nstart : dictionary, optional
        Starting value of Katz iteration for each node.
  
      normalized : bool, optional (default=True)
        If True normalize the resulting values.
  
      weight : None or string, optional (default=None)
        If None, all edge weights are considered equal.
        Otherwise holds the name of the edge attribute used as weight.
        In this measure the weight is interpreted as the connection strength.
  
      Returns
      -------
      nodes : dictionary
         Dictionary of nodes with Katz centrality as the value.
  
      Raises
      ------
      NetworkXError
         If the parameter `beta` is not a scalar but lacks a value for at least
         one node
  
      PowerIterationFailedConvergence
          If the algorithm fails to converge to the specified tolerance
          within the specified number of iterations of the power iteration
          method.
  
      Examples
      --------
      >>> import math
      >>> G = nx.path_graph(4)
      >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
      >>> centrality = nx.katz_centrality(G, 1 / phi - 0.01)
      >>> for n, c in sorted(centrality.items()):
      ...     print(f"{n} {c:.2f}")
      0 0.37
      1 0.60
      2 0.60
      3 0.37
  
      See Also
      --------
      katz_centrality_numpy
      eigenvector_centrality
      eigenvector_centrality_numpy
      :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
      :func:`~networkx.algorithms.link_analysis.hits_alg.hits`
  
      Notes
      -----
      Katz centrality was introduced by [2]_.
  
      This algorithm it uses the power method to find the eigenvector
      corresponding to the largest eigenvalue of the adjacency matrix of ``G``.
      The parameter ``alpha`` should be strictly less than the inverse of largest
      eigenvalue of the adjacency matrix for the algorithm to converge.
      You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
      eigenvalue of the adjacency matrix.
      The iteration will stop after ``max_iter`` iterations or an error tolerance of
      ``number_of_nodes(G) * tol`` has been reached.
  
      For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$,
      Katz centrality approaches the results for eigenvector centrality.
  
      For directed graphs this finds "left" eigenvectors which corresponds
      to the in-edges in the graph. For out-edges Katz centrality,
      first reverse the graph with ``G.reverse()``.
  
      References
      ----------
      .. [1] Mark E. J. Newman:
         Networks: An Introduction.
         Oxford University Press, USA, 2010, p. 720.
      .. [2] Leo Katz:
         A New Status Index Derived from Sociometric Index.
         Psychometrika 18(1):39–43, 1953
         https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
      """
      if len(G) == 0:
          return {}
  
      nnodes = G.number_of_nodes()
  
      if nstart is None:
          # choose starting vector with entries of 0
          x = {n: 0 for n in G}
      else:
          x = nstart
  
      try:
          b = dict.fromkeys(G, float(beta))
      except (TypeError, ValueError, AttributeError) as err:
          b = beta
          if set(beta) != set(G):
              raise nx.NetworkXError(
                  "beta dictionary must have a value for every node"
              ) from err
  
      # make up to max_iter iterations
      for _ in range(max_iter):
          xlast = x
          x = dict.fromkeys(xlast, 0)
          # do the multiplication y^T = Alpha * x^T A + Beta
          for n in x:
              for nbr in G[n]:
                  x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
          for n in x:
              x[n] = alpha * x[n] + b[n]
  
          # check convergence
          error = sum(abs(x[n] - xlast[n]) for n in x)
          if error < nnodes * tol:
              if normalized:
                  # normalize vector
                  try:
                      s = 1.0 / math.hypot(*x.values())
                  except ZeroDivisionError:
                      s = 1.0
              else:
                  s = 1
              for n in x:
                  x[n] *= s
              return x
      raise nx.PowerIterationFailedConvergence(max_iter)
  
imports: import math
  import networkx
  from networkx.utils import not_implemented_for
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, weight) and returns: dict, x