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# girvan_newman

girvan_newman(G, most_valuable_edge=None)

  Finds communities in a graph using the Girvan–Newman method.

  Parameters:
    G NetworkX graph
    most_valuable_edge function

      Function that takes a graph as input and outputs an edge. The edge returned by this function will be recomputed and removed at each iteration of the algorithm.

      If not specified, the edge with the highest `networkx.edge_betweenness_centrality()` will be used.

  Returns:
    iterator

      Iterator over tuples of sets of nodes in `G`. Each set of node is a community, each tuple is a sequence of communities at a particular level of the algorithm.

  Notes

  The Girvan–Newman algorithm detects communities by progressively removing edges from the original graph. The algorithm removes the “most valuable” edge, traditionally the edge with the highest betweenness centrality, at each step. As the graph breaks down into pieces, the tightly knit community structure is exposed and the result can be depicted as a dendrogram.

  Examples

  To get the first pair of communities:

      >>> G = nx.path_graph(10)
      >>> comp = nx.community.girvan_newman(G)
      >>> tuple(sorted(c) for c in next(comp))
      ([0, 1, 2, 3, 4], [5, 6, 7, 8, 9])

  To get only the first k tuples of communities, use `itertools.islice()`:

      >>> import itertools
      >>> G = nx.path_graph(8)
      >>> k = 2
      >>> comp = nx.community.girvan_newman(G)
      >>> for communities in itertools.islice(comp, k):
      ...     print(tuple(sorted(c) for c in communities))
      ...
      ([0, 1, 2, 3], [4, 5, 6, 7])
      ([0, 1], [2, 3], [4, 5, 6, 7])

  To stop getting tuples of communities once the number of communities is greater than k, use `itertools.takewhile()`:

      >>> import itertools
      >>> G = nx.path_graph(8)
      >>> k = 4
      >>> comp = nx.community.girvan_newman(G)
      >>> limited = itertools.takewhile(lambda c: len(c) <= k, comp)
      >>> for communities in limited:
      ...     print(tuple(sorted(c) for c in communities))
      ...
      ([0, 1, 2, 3], [4, 5, 6, 7])
      ([0, 1], [2, 3], [4, 5, 6, 7])
      ([0, 1], [2, 3], [4, 5], [6, 7])

  To just choose an edge to remove based on the weight:

      >>> from operator import itemgetter
      >>> G = nx.path_graph(10)
      >>> edges = G.edges()
      >>> nx.set_edge_attributes(G, {(u, v): v for u, v in edges}, "weight")
      >>> def heaviest(G):
      ...     u, v, w = max(G.edges(data="weight"), key=itemgetter(2))
      ...     return (u, v)
      ...
      >>> comp = nx.community.girvan_newman(G, most_valuable_edge=heaviest)
      >>> tuple(sorted(c) for c in next(comp))
      ([0, 1, 2, 3, 4, 5, 6, 7, 8], [9])

  To utilize edge weights when choosing an edge with, for example, the highest betweenness centrality:

      >>> from networkx import edge_betweenness_centrality as betweenness
      >>> def most_central_edge(G):
      ...     centrality = betweenness(G, weight="weight")
      ...     return max(centrality, key=centrality.get)
      ...
      >>> G = nx.path_graph(10)
      >>> comp = nx.community.girvan_newman(G, most_valuable_edge=most_central_edge)
      >>> tuple(sorted(c) for c in next(comp))
      ([0, 1, 2, 3, 4], [5, 6, 7, 8, 9])

  To specify a different ranking algorithm for edges, use the `most_valuable_edge` keyword argument:

      >>> from networkx import edge_betweenness_centrality
      >>> from random import random
      >>> def most_central_edge(G):
      ...     centrality = edge_betweenness_centrality(G)
      ...     max_cent = max(centrality.values())
      ...     # Scale the centrality values so they are between 0 and 1,
      ...     # and add some random noise.
      ...     centrality = {e: c / max_cent for e, c in centrality.items()}
      ...     # Add some random noise.
      ...     centrality = {e: c + random() for e, c in centrality.items()}
      ...     return max(centrality, key=centrality.get)
      ...
      >>> G = nx.path_graph(10)
      >>> comp = nx.community.girvan_newman(G, most_valuable_edge=most_central_edge)