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# Source code for networkx.algorithms.hierarchy

    """
    Flow Hierarchy.
    """

    import networkx as nx

    __all__ = ["flow_hierarchy"]

    @nx._dispatchable(edge_attrs="weight")
    def flow_hierarchy(G, weight=None):
        """Returns the flow hierarchy of a directed network.

        Flow hierarchy is defined as the fraction of edges not participating
        in cycles in a directed graph [1]_.

        Parameters
        ----------
        G : DiGraph or MultiDiGraph
           A directed graph

        weight : string, optional (default=None)
           Attribute to use for edge weights. If None the weight defaults to 1.

        Returns
        -------
        h : float
           Flow hierarchy value

        Raises
        ------
        NetworkXError
           If `G` is not a directed graph or if `G` has no edges.

        Notes
        -----
        The algorithm described in [1]_ computes the flow hierarchy through
        exponentiation of the adjacency matrix.  This function implements an
        alternative approach that finds strongly connected components.
        An edge is in a cycle if and only if it is in a strongly connected
        component, which can be found in $O(m)$ time using Tarjan's algorithm.

        References
        ----------
        .. [1] Luo, J.; Magee, C.L. (2011),
           Detecting evolving patterns of self-organizing networks by flow
           hierarchy measurement, Complexity, Volume 16 Issue 6 53-61.
           DOI: 10.1002/cplx.20368
           http://web.mit.edu/~cmagee/www/documents/28-DetectingEvolvingPatterns_FlowHierarchy.pdf
        """
        # corner case: G has no edges
        if nx.is_empty(G):
            raise nx.NetworkXError("flow_hierarchy not applicable to empty graphs")
        if not G.is_directed():
            raise nx.NetworkXError("G must be a digraph in flow_hierarchy")
        scc = nx.strongly_connected_components(G)
        return 1 - sum(G.subgraph(c).size(weight) for c in scc) / G.size(weight)