|
|
""" |
|
|
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 |
|
|
|
|
|
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) |
|
|
|
