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# simrank_similarity

simrank_similarity(G, source=None, target=None, importance_factor=0.9, max_iterations=1000, tolerance=0.0001)

  Returns the SimRank similarity of nodes in the graph `G`.

  SimRank is a similarity metric that says “two objects are considered to be similar if they are referenced by similar objects.” [1].

  The pseudo-code definition from the paper is:

      def simrank(G, u, v):
          in_neighbors_u = G.predecessors(u)
          in_neighbors_v = G.predecessors(v)
          scale = C / (len(in_neighbors_u) * len(in_neighbors_v))
          return scale * sum(
              simrank(G, w, x) for w, x in product(in_neighbors_u, in_neighbors_v)
          )

  where `G` is the graph, `u` is the source, `v` is the target, and `C` is a float decay or importance factor between 0 and 1.

  The SimRank algorithm for determining node similarity is defined in [2].

  Parameters:
    G NetworkX graph

      A NetworkX graph

    source node

      If this is specified, the returned dictionary maps each node `v` in the graph to the similarity between `source` and `v`.

    target node

      If both `source` and `target` are specified, the similarity value between `source` and `target` is returned. If `target` is specified but `source` is not, this argument is ignored.

    importance_factor float

      The relative importance of indirect neighbors with respect to direct neighbors.

    max_iterations integer

      Maximum number of iterations.

    tolerance float

      Error tolerance used to check convergence. When an iteration of the algorithm finds that no similarity value changes more than this amount, the algorithm halts.

  Returns:
    similarity dictionary or float

      If `source` and `target` are both `None`, this returns a dictionary of dictionaries, where keys are node pairs and value are similarity of the pair of nodes.

      If `source` is not `None` but `target` is, this returns a dictionary mapping node to the similarity of `source` and that node.

      If neither `source` nor `target` is `None`, this returns the similarity value for the given pair of nodes.

  Raises:
    ExceededMaxIterations

      If the algorithm does not converge within `max_iterations`.

    NodeNotFound

      If either `source` or `target` is not in `G`.

  References

  [1]

  https://en.wikipedia.org/wiki/SimRank

  [2]

  G. Jeh and J. Widom. “SimRank: a measure of structural-context similarity”, In KDD’02: Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 538–543. ACM Press, 2002.

  Examples

      >>> G = nx.cycle_graph(2)
      >>> nx.simrank_similarity(G)
      {0: {0: 1.0, 1: 0.0}, 1: {0: 0.0, 1: 1.0}}
      >>> nx.simrank_similarity(G, source=0)
      {0: 1.0, 1: 0.0}
      >>> nx.simrank_similarity(G, source=0, target=0)
      1.0

  The result of this function can be converted to a numpy array representing the SimRank matrix by using the node order of the graph to determine which row and column represent each node. Other ordering of nodes is also possible.

      >>> import numpy as np
      >>> sim = nx.simrank_similarity(G)
      >>> np.array([[sim[u][v] for v in G] for u in G])
      array([[1., 0.],
             [0., 1.]])
      >>> sim_1d = nx.simrank_similarity(G, source=0)
      >>> np.array([sim[0][v] for v in G])
      array([1., 0.])