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"""
Flow Hierarchy.
"""
import networkx as nx
__all__ = ["flow_hierarchy"]
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 : key,optional (default=None)
Attribute to use for node weights. If None the weight defaults to 1.
Returns
-------
h : float
Flow hierarchy value
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
"""
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)
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