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flow_hierarchy(G, weight=None)[source]

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].

  • 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.

h – Flow heirarchy value

Return type:



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.


[1](1, 2) 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