import functools
import networkx as nx
__all__ = [
"edge_betweenness_partition",
"edge_current_flow_betweenness_partition",
]
[docs]
@nx._dispatchable(edge_attrs="weight")
def edge_betweenness_partition(G, number_of_sets, *, weight=None):
"""Partition created by iteratively removing the highest edge betweenness edge.
This algorithm works by calculating the edge betweenness for all
edges and removing the edge with the highest value. It is then
determined whether the graph has been broken into at least
`number_of_sets` connected components.
If not the process is repeated.
Parameters
----------
G : NetworkX Graph, DiGraph or MultiGraph
Graph to be partitioned
number_of_sets : int
Number of sets in the desired partition of the graph
weight : key, optional, default=None
The key to use if using weights for edge betweenness calculation
Returns
-------
C : list of sets
Partition of the nodes of G
Raises
------
NetworkXError
If number_of_sets is <= 0 or if number_of_sets > len(G)
Examples
--------
>>> G = nx.karate_club_graph()
>>> part = nx.community.edge_betweenness_partition(G, 2)
>>> {0, 1, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 19, 21} in part
True
>>> {2, 8, 9, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} in part
True
See Also
--------
edge_current_flow_betweenness_partition
Notes
-----
This algorithm is fairly slow, as both the calculation of connected
components and edge betweenness relies on all pairs shortest
path algorithms. They could potentially be combined to cut down
on overall computation time.
References
----------
.. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
Volume 486, Issue 3-5 p. 75-174
http://arxiv.org/abs/0906.0612
"""
if number_of_sets <= 0:
raise nx.NetworkXError("number_of_sets must be >0")
if number_of_sets == 1:
return [set(G)]
if number_of_sets == len(G):
return [{n} for n in G]
if number_of_sets > len(G):
raise nx.NetworkXError("number_of_sets must be <= len(G)")
H = G.copy()
partition = list(nx.connected_components(H))
while len(partition) < number_of_sets:
ranking = nx.edge_betweenness_centrality(H, weight=weight)
edge = max(ranking, key=ranking.get)
H.remove_edge(*edge)
partition = list(nx.connected_components(H))
return partition
[docs]
@nx._dispatchable(edge_attrs="weight")
def edge_current_flow_betweenness_partition(G, number_of_sets, *, weight=None):
"""Partition created by removing the highest edge current flow betweenness edge.
This algorithm works by calculating the edge current flow
betweenness for all edges and removing the edge with the
highest value. It is then determined whether the graph has
been broken into at least `number_of_sets` connected
components. If not the process is repeated.
Parameters
----------
G : NetworkX Graph, DiGraph or MultiGraph
Graph to be partitioned
number_of_sets : int
Number of sets in the desired partition of the graph
weight : key, optional (default=None)
The edge attribute key to use as weights for
edge current flow betweenness calculations
Returns
-------
C : list of sets
Partition of G
Raises
------
NetworkXError
If number_of_sets is <= 0 or number_of_sets > len(G)
Examples
--------
>>> G = nx.karate_club_graph()
>>> part = nx.community.edge_current_flow_betweenness_partition(G, 2)
>>> {0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 19, 21} in part
True
>>> {8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} in part
True
See Also
--------
edge_betweenness_partition
Notes
-----
This algorithm is extremely slow, as the recalculation of the edge
current flow betweenness is extremely slow.
References
----------
.. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
Volume 486, Issue 3-5 p. 75-174
http://arxiv.org/abs/0906.0612
"""
if number_of_sets <= 0:
raise nx.NetworkXError("number_of_sets must be >0")
elif number_of_sets == 1:
return [set(G)]
elif number_of_sets == len(G):
return [{n} for n in G]
elif number_of_sets > len(G):
raise nx.NetworkXError("number_of_sets must be <= len(G)")
rank = functools.partial(
nx.edge_current_flow_betweenness_centrality, normalized=False, weight=weight
)
# current flow requires a connected network so we track the components explicitly
H = G.copy()
partition = list(nx.connected_components(H))
if len(partition) > 1:
Hcc_subgraphs = [H.subgraph(cc).copy() for cc in partition]
else:
Hcc_subgraphs = [H]
ranking = {}
for Hcc in Hcc_subgraphs:
ranking.update(rank(Hcc))
while len(partition) < number_of_sets:
edge = max(ranking, key=ranking.get)
for cc, Hcc in zip(partition, Hcc_subgraphs):
if edge[0] in cc:
Hcc.remove_edge(*edge)
del ranking[edge]
splitcc_list = list(nx.connected_components(Hcc))
if len(splitcc_list) > 1:
# there are 2 connected components. split off smaller one
cc_new = min(splitcc_list, key=len)
Hcc_new = Hcc.subgraph(cc_new).copy()
# update edge rankings for Hcc_new
newranks = rank(Hcc_new)
for e, r in newranks.items():
ranking[e if e in ranking else e[::-1]] = r
# append new cc and Hcc to their lists.
partition.append(cc_new)
Hcc_subgraphs.append(Hcc_new)
# leave existing cc and Hcc in their lists, but shrink them
Hcc.remove_nodes_from(cc_new)
cc.difference_update(cc_new)
# update edge rankings for Hcc whether it was split or not
newranks = rank(Hcc)
for e, r in newranks.items():
ranking[e if e in ranking else e[::-1]] = r
break
return partition
