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- """
- 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)
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