""" Fast approximation for node connectivity
"""
# Copyright (C) 2015 by
# Jordi Torrents <jtorrents@milnou.net>
# All rights reserved.
# BSD license.
import itertools
from operator import itemgetter
import networkx as nx
__author__ = """\n""".join(['Jordi Torrents <jtorrents@milnou.net>'])
__all__ = ['local_node_connectivity',
'node_connectivity',
'all_pairs_node_connectivity']
INF = float('inf')
[docs]def local_node_connectivity(G, source, target, cutoff=None):
"""Compute node connectivity between source and target.
Pairwise or local node connectivity between two distinct and nonadjacent
nodes is the minimum number of nodes that must be removed (minimum
separating cutset) to disconnect them. By Menger's theorem, this is equal
to the number of node independent paths (paths that share no nodes other
than source and target). Which is what we compute in this function.
This algorithm is a fast approximation that gives an strict lower
bound on the actual number of node independent paths between two nodes [1]_.
It works for both directed and undirected graphs.
Parameters
----------
G : NetworkX graph
source : node
Starting node for node connectivity
target : node
Ending node for node connectivity
cutoff : integer
Maximum node connectivity to consider. If None, the minimum degree
of source or target is used as a cutoff. Default value None.
Returns
-------
k: integer
pairwise node connectivity
Examples
--------
>>> # Platonic octahedral graph has node connectivity 4
>>> # for each non adjacent node pair
>>> from networkx.algorithms import approximation as approx
>>> G = nx.octahedral_graph()
>>> approx.local_node_connectivity(G, 0, 5)
4
Notes
-----
This algorithm [1]_ finds node independents paths between two nodes by
computing their shortest path using BFS, marking the nodes of the path
found as 'used' and then searching other shortest paths excluding the
nodes marked as used until no more paths exist. It is not exact because
a shortest path could use nodes that, if the path were longer, may belong
to two different node independent paths. Thus it only guarantees an
strict lower bound on node connectivity.
Note that the authors propose a further refinement, losing accuracy and
gaining speed, which is not implemented yet.
See also
--------
all_pairs_node_connectivity
node_connectivity
References
----------
.. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
http://eclectic.ss.uci.edu/~drwhite/working.pdf
"""
if target == source:
raise nx.NetworkXError("source and target have to be different nodes.")
# Maximum possible node independent paths
if G.is_directed():
possible = min(G.out_degree(source), G.in_degree(target))
else:
possible = min(G.degree(source), G.degree(target))
K = 0
if not possible:
return K
if cutoff is None:
cutoff = INF
exclude = set()
for i in range(min(possible, cutoff)):
try:
path = _bidirectional_shortest_path(G, source, target, exclude)
exclude.update(set(path))
K += 1
except nx.NetworkXNoPath:
break
return K
[docs]def node_connectivity(G, s=None, t=None):
r"""Returns an approximation for node connectivity for a graph or digraph G.
Node connectivity is equal to the minimum number of nodes that
must be removed to disconnect G or render it trivial. By Menger's theorem,
this is equal to the number of node independent paths (paths that
share no nodes other than source and target).
If source and target nodes are provided, this function returns the
local node connectivity: the minimum number of nodes that must be
removed to break all paths from source to target in G.
This algorithm is based on a fast approximation that gives an strict lower
bound on the actual number of node independent paths between two nodes [1]_.
It works for both directed and undirected graphs.
Parameters
----------
G : NetworkX graph
Undirected graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
Returns
-------
K : integer
Node connectivity of G, or local node connectivity if source
and target are provided.
Examples
--------
>>> # Platonic octahedral graph is 4-node-connected
>>> from networkx.algorithms import approximation as approx
>>> G = nx.octahedral_graph()
>>> approx.node_connectivity(G)
4
Notes
-----
This algorithm [1]_ finds node independents paths between two nodes by
computing their shortest path using BFS, marking the nodes of the path
found as 'used' and then searching other shortest paths excluding the
nodes marked as used until no more paths exist. It is not exact because
a shortest path could use nodes that, if the path were longer, may belong
to two different node independent paths. Thus it only guarantees an
strict lower bound on node connectivity.
See also
--------
all_pairs_node_connectivity
local_node_connectivity
References
----------
.. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
http://eclectic.ss.uci.edu/~drwhite/working.pdf
"""
if (s is not None and t is None) or (s is None and t is not None):
raise nx.NetworkXError('Both source and target must be specified.')
# Local node connectivity
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
return local_node_connectivity(G, s, t)
# Global node connectivity
if G.is_directed():
connected_func = nx.is_weakly_connected
iter_func = itertools.permutations
def neighbors(v):
return itertools.chain(G.predecessors(v), G.successors(v))
else:
connected_func = nx.is_connected
iter_func = itertools.combinations
neighbors = G.neighbors
if not connected_func(G):
return 0
# Choose a node with minimum degree
v, minimum_degree = min(G.degree(), key=itemgetter(1))
# Node connectivity is bounded by minimum degree
K = minimum_degree
# compute local node connectivity with all non-neighbors nodes
# and store the minimum
for w in set(G) - set(neighbors(v)) - set([v]):
K = min(K, local_node_connectivity(G, v, w, cutoff=K))
# Same for non adjacent pairs of neighbors of v
for x, y in iter_func(neighbors(v), 2):
if y not in G[x] and x != y:
K = min(K, local_node_connectivity(G, x, y, cutoff=K))
return K
[docs]def all_pairs_node_connectivity(G, nbunch=None, cutoff=None):
""" Compute node connectivity between all pairs of nodes.
Pairwise or local node connectivity between two distinct and nonadjacent
nodes is the minimum number of nodes that must be removed (minimum
separating cutset) to disconnect them. By Menger's theorem, this is equal
to the number of node independent paths (paths that share no nodes other
than source and target). Which is what we compute in this function.
This algorithm is a fast approximation that gives an strict lower
bound on the actual number of node independent paths between two nodes [1]_.
It works for both directed and undirected graphs.
Parameters
----------
G : NetworkX graph
nbunch: container
Container of nodes. If provided node connectivity will be computed
only over pairs of nodes in nbunch.
cutoff : integer
Maximum node connectivity to consider. If None, the minimum degree
of source or target is used as a cutoff in each pair of nodes.
Default value None.
Returns
-------
K : dictionary
Dictionary, keyed by source and target, of pairwise node connectivity
See Also
--------
local_node_connectivity
all_pairs_node_connectivity
References
----------
.. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
http://eclectic.ss.uci.edu/~drwhite/working.pdf
"""
if nbunch is None:
nbunch = G
else:
nbunch = set(nbunch)
directed = G.is_directed()
if directed:
iter_func = itertools.permutations
else:
iter_func = itertools.combinations
all_pairs = {n: {} for n in nbunch}
for u, v in iter_func(nbunch, 2):
k = local_node_connectivity(G, u, v, cutoff=cutoff)
all_pairs[u][v] = k
if not directed:
all_pairs[v][u] = k
return all_pairs
def _bidirectional_shortest_path(G, source, target, exclude):
"""Return shortest path between source and target ignoring nodes in the
container 'exclude'.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
exclude: container
Container for nodes to exclude from the search for shortest paths
Returns
-------
path: list
Shortest path between source and target ignoring nodes in 'exclude'
Raises
------
NetworkXNoPath: exception
If there is no path or if nodes are adjacent and have only one path
between them
Notes
-----
This function and its helper are originally from
networkx.algorithms.shortest_paths.unweighted and are modified to
accept the extra parameter 'exclude', which is a container for nodes
already used in other paths that should be ignored.
References
----------
.. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
http://eclectic.ss.uci.edu/~drwhite/working.pdf
"""
# call helper to do the real work
results = _bidirectional_pred_succ(G, source, target, exclude)
pred, succ, w = results
# build path from pred+w+succ
path = []
# from source to w
while w is not None:
path.append(w)
w = pred[w]
path.reverse()
# from w to target
w = succ[path[-1]]
while w is not None:
path.append(w)
w = succ[w]
return path
def _bidirectional_pred_succ(G, source, target, exclude):
# does BFS from both source and target and meets in the middle
# excludes nodes in the container "exclude" from the search
if source is None or target is None:
raise nx.NetworkXException(
"Bidirectional shortest path called without source or target")
if target == source:
return ({target: None}, {source: None}, source)
# handle either directed or undirected
if G.is_directed():
Gpred = G.predecessors
Gsucc = G.successors
else:
Gpred = G.neighbors
Gsucc = G.neighbors
# predecesssor and successors in search
pred = {source: None}
succ = {target: None}
# initialize fringes, start with forward
forward_fringe = [source]
reverse_fringe = [target]
level = 0
while forward_fringe and reverse_fringe:
# Make sure that we iterate one step forward and one step backwards
# thus source and target will only trigger "found path" when they are
# adjacent and then they can be safely included in the container 'exclude'
level += 1
if not level % 2 == 0:
this_level = forward_fringe
forward_fringe = []
for v in this_level:
for w in Gsucc(v):
if w in exclude:
continue
if w not in pred:
forward_fringe.append(w)
pred[w] = v
if w in succ:
return pred, succ, w # found path
else:
this_level = reverse_fringe
reverse_fringe = []
for v in this_level:
for w in Gpred(v):
if w in exclude:
continue
if w not in succ:
succ[w] = v
reverse_fringe.append(w)
if w in pred:
return pred, succ, w # found path
raise nx.NetworkXNoPath("No path between %s and %s." % (source, target))