def test_gl1(self):
G = read_graph("gl1")
s = 1
t = len(G)
R = build_residual_network(G, "capacity")
kwargs = dict(residual=R)
for flow_func in flow_funcs:
validate_flows(G, s, t, 156545, flow_func(G, s, t, **kwargs), flow_func)
def test_wlm3(self):
G = read_graph("wlm3")
s = 1
t = len(G)
R = build_residual_network(G, "capacity")
kwargs = dict(residual=R)
for flow_func in flow_funcs:
validate_flows(G, s, t, 11875108, flow_func(G, s, t, **kwargs), flow_func)
def test_gw1(self):
G = read_graph('gw1')
s = 1
t = len(G)
R = build_residual_network(G, 'capacity')
kwargs = dict(residual=R)
for flow_func in flow_funcs:
validate_flows(G, s, t, 1202018, flow_func(G, s, t, **kwargs),
flow_func)
def test_complete_graph(self):
N = 50
G = nx.complete_graph(N)
nx.set_edge_attributes(G, "capacity", 5)
R = build_residual_network(G, "capacity")
kwargs = dict(residual=R)
for flow_func in flow_funcs:
kwargs["flow_func"] = flow_func
flow_value = nx.maximum_flow_value(G, 1, 2, **kwargs)
assert_equal(flow_value, 5 * (N - 1), msg=msg.format(flow_func.__name__))
def test_pyramid(self):
N = 10
# N = 100 # this gives a graph with 5051 nodes
G = gen_pyramid(N)
R = build_residual_network(G, "capacity")
kwargs = dict(residual=R)
for flow_func in flow_funcs:
kwargs["flow_func"] = flow_func
flow_value = nx.maximum_flow_value(G, (0, 0), "t", **kwargs)
assert_almost_equal(flow_value, 1.0, msg=msg.format(flow_func.__name__))
def test_wlm3(self):
G = read_graph('wlm3')
s = 1
t = len(G)
R = build_residual_network(G, 'capacity')
kwargs = dict(residual=R)
# do one flow_func to save time
flow_func = flow_funcs[0]
validate_flows(G, s, t, 11875108, flow_func(G, s, t, **kwargs),
flow_func)
def test_reusing_residual(self):
G = self.G
fv = 3.0
s, t = 'x', 'y'
R = build_residual_network(G, 'capacity')
for interface_func in interface_funcs:
for flow_func in flow_funcs:
for i in range(3):
result = interface_func(G, 'x', 'y', flow_func=flow_func,
residual=R)
if interface_func in max_min_funcs:
result = result[0]
assert_equal(fv, result,
msg=msgi.format(flow_func.__name__,
interface_func.__name__))
def average_node_connectivity(G, flow_func=None):
r"""Returns the average connectivity of a graph G.
The average connectivity `\bar{\kappa}` of a graph G is the average
of local node connectivity over all pairs of nodes of G [1]_ .
.. math::
\bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}
Parameters
----------
G : NetworkX graph
Undirected graph
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity`
for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns
-------
K : float
Average node connectivity
See also
--------
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average
connectivity of a graph. Discrete mathematics 252(1-3), 31-45.
http://www.sciencedirect.com/science/article/pii/S0012365X01001807
"""
if G.is_directed():
iter_func = itertools.permutations
else:
iter_func = itertools.combinations
# Reuse the auxiliary digraph and the residual network
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
num, den = 0, 0
for u, v in iter_func(G, 2):
num += local_node_connectivity(G, u, v, **kwargs)
den += 1
if den == 0: # Null Graph
return 0
return num / den
def node_connectivity(G, s=None, t=None, flow_func=None):
r"""Returns 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. 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.
Parameters
----------
G : NetworkX graph
Undirected graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
K : integer
Node connectivity of G, or local node connectivity if source
and target are provided.
Examples
--------
>>> # Platonic icosahedral graph is 5-node-connected
>>> G = nx.icosahedral_graph()
>>> nx.node_connectivity(G)
5
You can use alternative flow algorithms for the underlying maximum
flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local node connectivity.
>>> nx.node_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_node_connectivity` for details.
Notes
-----
This is a flow based implementation of node connectivity. The
algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2))`
maximum flow problems on an auxiliary digraph. Where `\delta`
is the minimum degree of G. For details about the auxiliary
digraph and the computation of local node connectivity see
:meth:`local_node_connectivity`. This implementation is based
on algorithm 11 in [1]_.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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, flow_func=flow_func)
# Global node connectivity
if G.is_directed():
if not nx.is_weakly_connected(G):
return 0
iter_func = itertools.permutations
# It is necessary to consider both predecessors
# and successors for directed graphs
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors_iter(v),
G.successors_iter(v)])
else:
if not nx.is_connected(G):
return 0
iter_func = itertools.combinations
neighbors = G.neighbors_iter
# Reuse the auxiliary digraph and the residual network
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
# Pick a node with minimum degree
degree = G.degree()
minimum_degree = min(degree.values())
v = next(n for n, d in degree.items() if d == minimum_degree)
# Node connectivity is bounded by degree.
K = minimum_degree
# compute local node connectivity with all its non-neighbors nodes
for w in set(G) - set(neighbors(v)) - set([v]):
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, v, w, **kwargs))
# Also for non adjacent pairs of neighbors of v
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, x, y, **kwargs))
return K
def all_node_cuts(G, k=None, flow_func=None):
r"""Returns all minimum k cutsets of an undirected graph G.
This implementation is based on Kanevsky's algorithm [1]_ for finding all
minimum-size node cut-sets of an undirected graph G; ie the set (or sets)
of nodes of cardinality equal to the node connectivity of G. Thus if
removed, would break G into two or more connected components.
Parameters
----------
G : NetworkX graph
Undirected graph
k : Integer
Node connectivity of the input graph. If k is None, then it is
computed. Default value: None.
flow_func : function
Function to perform the underlying flow computations. Default value
edmonds_karp. This function performs better in sparse graphs with
right tailed degree distributions. shortest_augmenting_path will
perform better in denser graphs.
Returns
-------
cuts : a generator of node cutsets
Each node cutset has cardinality equal to the node connectivity of
the input graph.
Examples
--------
>>> # A two-dimensional grid graph has 4 cutsets of cardinality 2
>>> G = nx.grid_2d_graph(5, 5)
>>> cutsets = list(nx.all_node_cuts(G))
>>> len(cutsets)
4
>>> all(2 == len(cutset) for cutset in cutsets)
True
>>> nx.node_connectivity(G)
2
Notes
-----
This implementation is based on the sequential algorithm for finding all
minimum-size separating vertex sets in a graph [1]_. The main idea is to
compute minimum cuts using local maximum flow computations among a set
of nodes of highest degree and all other non-adjacent nodes in the Graph.
Once we find a minimum cut, we add an edge between the high degree
node and the target node of the local maximum flow computation to make
sure that we will not find that minimum cut again.
See also
--------
node_connectivity
edmonds_karp
shortest_augmenting_path
References
----------
.. [1] Kanevsky, A. (1993). Finding all minimum-size separating vertex
sets in a graph. Networks 23(6), 533--541.
http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract
"""
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is disconnected.')
# Address some corner cases first.
# For cycle graphs
if G.order() == G.size():
if all(2 == d for n, d in G.degree()):
seen = set()
for u in G:
for v in nx.non_neighbors(G, u):
if (u, v) not in seen and (v, u) not in seen:
yield {v, u}
seen.add((v, u))
return
# For complete Graphs
if nx.density(G) == 1:
for cut_set in combinations(G, len(G) - 1):
yield set(cut_set)
return
# Initialize data structures.
# Keep track of the cuts already computed so we do not repeat them.
seen = []
# Even-Tarjan reduction is what we call auxiliary digraph
# for node connectivity.
H = build_auxiliary_node_connectivity(G)
mapping = H.graph['mapping']
R = build_residual_network(H, 'capacity')
kwargs = dict(capacity='capacity', residual=R)
# Define default flow function
if flow_func is None:
flow_func = default_flow_func
if flow_func is shortest_augmenting_path:
kwargs['two_phase'] = True
# Begin the actual algorithm
# step 1: Find node connectivity k of G
if k is None:
k = nx.node_connectivity(G, flow_func=flow_func)
# step 2:
# Find k nodes with top degree, call it X:
X = {n for n, d in sorted(G.degree(), key=itemgetter(1), reverse=True)[:k]}
# Check if X is a k-node-cutset
if _is_separating_set(G, X):
seen.append(X)
yield X
for x in X:
# step 3: Compute local connectivity flow of x with all other
# non adjacent nodes in G
non_adjacent = set(G) - X - set(G[x])
for v in non_adjacent:
# step 4: compute maximum flow in an Even-Tarjan reduction H of G
# and step:5 build the associated residual network R
R = flow_func(H, '%sB' % mapping[x], '%sA' % mapping[v], **kwargs)
flow_value = R.graph['flow_value']
if flow_value == k:
# Remove saturated edges form the residual network
saturated_edges = [(u, w, d) for (u, w, d) in
R.edges(data=True)
if d['capacity'] == d['flow']]
R.remove_edges_from(saturated_edges)
# step 6: shrink the strongly connected components of
# residual flow network R and call it L
L = nx.condensation(R)
cmap = L.graph['mapping']
# step 7: Compute antichains of L; they map to closed sets in H
# Any edge in H that links a closed set is part of a cutset
for antichain in nx.antichains(L):
# Nodes in an antichain of the condensation graph of
# the residual network map to a closed set of nodes that
# define a node partition of the auxiliary digraph H.
S = {n for n, scc in cmap.items() if scc in antichain}
# Find the cutset that links the node partition (S,~S) in H
cutset = set()
for u in S:
cutset.update((u, w) for w in H[u] if w not in S)
# The edges in H that form the cutset are internal edges
# (ie edges that represent a node of the original graph G)
node_cut = {H.nodes[n]['id'] for edge in cutset for n in edge}
if len(node_cut) == k:
if node_cut not in seen:
yield node_cut
seen.append(node_cut)
# Add an edge (x, v) to make sure that we do not
# find this cutset again. This is equivalent
# of adding the edge in the input graph
# G.add_edge(x, v) and then regenerate H and R:
# Add edges to the auxiliary digraph.
H.add_edge('%sB' % mapping[x], '%sA' % mapping[v],
capacity=1)
H.add_edge('%sB' % mapping[v], '%sA' % mapping[x],
capacity=1)
# Add edges to the residual network.
R.add_edge('%sB' % mapping[x], '%sA' % mapping[v],
capacity=1)
R.add_edge('%sA' % mapping[v], '%sB' % mapping[x],
capacity=1)
break
# Add again the saturated edges to reuse the residual network
R.add_edges_from(saturated_edges)
def all_pairs_node_connectivity(G, nbunch=None, flow_func=None):
"""Compute node connectivity between all pairs of nodes of G.
Parameters
----------
G : NetworkX graph
Undirected graph
nbunch: container
Container of nodes. If provided node connectivity will be computed
only over pairs of nodes in nbunch.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
all_pairs : dict
A dictionary with node connectivity between all pairs of nodes
in G, or in nbunch if provided.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`local_edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
"""
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}
# Reuse auxiliary digraph and residual network
H = build_auxiliary_node_connectivity(G)
mapping = H.graph['mapping']
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
for u, v in iter_func(nbunch, 2):
K = local_node_connectivity(G, u, v, **kwargs)
all_pairs[u][v] = K
if not directed:
all_pairs[v][u] = K
return all_pairs
def minimum_edge_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of edges of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of edges of minimum cardinality that, if removed, would break
all paths among source and target in G. If not, it returns a set of
edges of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
cutset : set
Set of edges that, if removed, would disconnect G. If source
and target nodes are provided, the set contians the edges that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has edge connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_edge_cut(G))
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of minimum edge cut. For
undirected graphs the algorithm works by finding a 'small' dominating
set of nodes of G (see algorithm 7 in [1]_) and computing the maximum
flow between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_. For
directed graphs, the algorithm does n calls to the max flow function.
The function raises an error if the directed graph is not weakly
connected and returns an empty set if it is weakly connected.
It is an implementation of algorithm 8 in [1]_.
See also
--------
:meth:`minimum_st_edge_cut`
:meth:`minimum_node_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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.")
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, "capacity")
kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)
# Local minimum edge cut if s and t are not None
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 minimum_st_edge_cut(H, s, t, **kwargs)
# Global minimum edge cut
# Analog to the algoritm for global edge connectivity
if G.is_directed():
# Based on algorithm 8 in [1]
if not nx.is_weakly_connected(G):
raise nx.NetworkXError("Input graph is not connected")
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
nodes = list(G)
n = len(nodes)
for i in range(n):
try:
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1], **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
except IndexError: # Last node!
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
else: # undirected
# Based on algorithm 6 in [1]
if not nx.is_connected(G):
raise nx.NetworkXError("Input graph is not connected")
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating set will always be of one node
# thus we return min_cut, which now contains the edges of a node
# with minimum degree
return min_cut
for w in D:
this_cut = minimum_st_edge_cut(H, v, w, **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
def minimum_node_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of nodes of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of nodes of minimum cardinality that, if removed, would destroy
all paths among source and target in G. If not, it returns a set
of nodes of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
cutset : set
Set of nodes that, if removed, would disconnect G. If source
and target nodes are provided, the set contians the nodes that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has node connectivity 5
>>> G = nx.icosahedral_graph()
>>> node_cut = nx.minimum_node_cut(G)
>>> len(node_cut)
5
You can use alternative flow algorithms for the underlying maximum
flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
True
If you specify a pair of nodes (source and target) as parameters,
this function returns a local st node cut.
>>> len(nx.minimum_node_cut(G, 3, 7))
5
If you need to perform several local st cuts among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`minimum_st_node_cut` for details.
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of maximum flow computations to determine
the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed
and undirected graphs. This implementation is based on algorithm 11
in [1]_.
See also
--------
:meth:`minimum_st_node_cut`
:meth:`minimum_cut`
:meth:`minimum_edge_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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 minimum node cut.
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 minimum_st_node_cut(G, s, t, flow_func=flow_func)
# Global minimum node cut.
# Analog to the algoritm 11 for global node connectivity in [1].
if G.is_directed():
if not nx.is_weakly_connected(G):
raise nx.NetworkXError("Input graph is not connected")
iter_func = itertools.permutations
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])
else:
if not nx.is_connected(G):
raise nx.NetworkXError("Input graph is not connected")
iter_func = itertools.combinations
neighbors = G.neighbors
# Reuse the auxiliary digraph and the residual network.
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, "capacity")
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
# Choose a node with minimum degree.
v = min(G, key=G.degree)
# Initial node cutset is all neighbors of the node with minimum degree.
min_cut = set(G[v])
# Compute st node cuts between v and all its non-neighbors nodes in G.
for w in set(G) - set(neighbors(v)) - set([v]):
this_cut = minimum_st_node_cut(G, v, w, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
# Also for non adjacent pairs of neighbors of v.
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
this_cut = minimum_st_node_cut(G, x, y, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
return min_cut
def edge_connectivity(G, s=None, t=None, flow_func=None):
r"""Returns the edge connectivity of the graph or digraph G.
The edge connectivity is equal to the minimum number of edges that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local edge
connectivity: the minimum number of edges that must be removed to
break all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected or directed graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
K : integer
Edge connectivity for G, or local edge connectivity if source
and target were provided
Examples
--------
>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(G)
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of global edge connectivity.
For undirected graphs the algorithm works by finding a 'small'
dominating set of nodes of G (see algorithm 7 in [1]_ ) and
computing local maximum flow (see :meth:`local_edge_connectivity`)
between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_ .
For directed graphs, the algorithm does n calls to the maximum
flow function. This is an implementation of algorithm 8 in [1]_ .
See also
--------
:meth:`local_edge_connectivity`
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
:meth:`k_edge_components`
:meth:`k_edge_subgraphs`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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 edge 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_edge_connectivity(G, s, t, flow_func=flow_func)
# Global edge connectivity
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
if G.is_directed():
# Algorithm 8 in [1]
if not nx.is_weakly_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
nodes = list(G)
n = len(nodes)
for i in range(n):
kwargs['cutoff'] = L
try:
L = min(
L,
local_edge_connectivity(G, nodes[i], nodes[i + 1],
**kwargs))
except IndexError: # last node!
L = min(
L, local_edge_connectivity(G, nodes[i], nodes[0],
**kwargs))
return L
else: # undirected
# Algorithm 6 in [1]
if not nx.is_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating sets will always be of one node
# thus we return min degree
return L
for w in D:
kwargs['cutoff'] = L
L = min(L, local_edge_connectivity(G, v, w, **kwargs))
return L
def average_node_connectivity(G, flow_func=None):
r"""Returns the average connectivity of a graph G.
The average connectivity `\bar{\kappa}` of a graph G is the average
of local node connectivity over all pairs of nodes of G [1]_ .
.. math::
\bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}
Parameters
----------
G : NetworkX graph
Undirected graph
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity`
for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns
-------
K : float
Average node connectivity
See also
--------
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average
connectivity of a graph. Discrete mathematics 252(1-3), 31-45.
http://www.sciencedirect.com/science/article/pii/S0012365X01001807
"""
if G.is_directed():
iter_func = itertools.permutations
else:
iter_func = itertools.combinations
# Reuse the auxiliary digraph and the residual network
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
num, den = 0, 0
for u, v in iter_func(G, 2):
num += local_node_connectivity(G, u, v, **kwargs)
den += 1
if den == 0: # Null Graph
return 0
return num / den
def all_pairs_node_connectivity(G, nbunch=None, flow_func=None):
"""Compute node connectivity between all pairs of nodes of G.
Parameters
----------
G : NetworkX graph
Undirected graph
nbunch: container
Container of nodes. If provided node connectivity will be computed
only over pairs of nodes in nbunch.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
all_pairs : dict
A dictionary with node connectivity between all pairs of nodes
in G, or in nbunch if provided.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`local_edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
"""
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}
# Reuse auxiliary digraph and residual network
H = build_auxiliary_node_connectivity(G)
mapping = H.graph['mapping']
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
for u, v in iter_func(nbunch, 2):
K = local_node_connectivity(G, u, v, **kwargs)
all_pairs[u][v] = K
if not directed:
all_pairs[v][u] = K
return all_pairs
def minimum_node_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of nodes of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of nodes of minimum cardinality that, if removed, would destroy
all paths among source and target in G. If not, it returns a set
of nodes of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
cutset : set
Set of nodes that, if removed, would disconnect G. If source
and target nodes are provided, the set contians the nodes that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has node connectivity 5
>>> G = nx.icosahedral_graph()
>>> node_cut = nx.minimum_node_cut(G)
>>> len(node_cut)
5
You can use alternative flow algorithms for the underlying maximum
flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
True
If you specify a pair of nodes (source and target) as parameters,
this function returns a local st node cut.
>>> len(nx.minimum_node_cut(G, 3, 7))
5
If you need to perform several local st cuts among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`minimum_st_node_cut` for details.
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of maximum flow computations to determine
the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed
and undirected graphs. This implementation is based on algorithm 11
in [1]_.
See also
--------
:meth:`minimum_st_node_cut`
:meth:`minimum_cut`
:meth:`minimum_edge_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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 minimum node cut.
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 minimum_st_node_cut(G, s, t, flow_func=flow_func)
# Global minimum node cut.
# Analog to the algoritm 11 for global node connectivity in [1].
if G.is_directed():
if not nx.is_weakly_connected(G):
raise nx.NetworkXError('Input graph is not connected')
iter_func = itertools.permutations
def neighbors(v):
return itertools.chain.from_iterable(
[G.predecessors(v), G.successors(v)])
else:
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is not connected')
iter_func = itertools.combinations
neighbors = G.neighbors
# Reuse the auxiliary digraph and the residual network.
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
# Choose a node with minimum degree.
v = min(G, key=G.degree)
# Initial node cutset is all neighbors of the node with minimum degree.
min_cut = set(G[v])
# Compute st node cuts between v and all its non-neighbors nodes in G.
for w in set(G) - set(neighbors(v)) - set([v]):
this_cut = minimum_st_node_cut(G, v, w, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
# Also for non adjacent pairs of neighbors of v.
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
this_cut = minimum_st_node_cut(G, x, y, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
return min_cut
def all_node_cuts(G, k=None, flow_func=None):
r"""Returns all minimum k cutsets of an undirected graph G.
This implementation is based on Kanevsky's algorithm [1]_ for finding all
minimum-size node cut-sets of an undirected graph G; ie the set (or sets)
of nodes of cardinality equal to the node connectivity of G. Thus if
removed, would break G into two or more connected components.
Parameters
----------
G : NetworkX graph
Undirected graph
k : Integer
Node connectivity of the input graph. If k is None, then it is
computed. Default value: None.
flow_func : function
Function to perform the underlying flow computations. Default value
edmonds_karp. This function performs better in sparse graphs with
right tailed degree distributions. shortest_augmenting_path will
perform better in denser graphs.
Returns
-------
cuts : a generator of node cutsets
Each node cutset has cardinality equal to the node connectivity of
the input graph.
Examples
--------
>>> # A two-dimensional grid graph has 4 cutsets of cardinality 2
>>> G = nx.grid_2d_graph(5, 5)
>>> cutsets = list(nx.all_node_cuts(G))
>>> len(cutsets)
4
>>> all(2 == len(cutset) for cutset in cutsets)
True
>>> nx.node_connectivity(G)
2
Notes
-----
This implementation is based on the sequential algorithm for finding all
minimum-size separating vertex sets in a graph [1]_. The main idea is to
compute minimum cuts using local maximum flow computations among a set
of nodes of highest degree and all other non-adjacent nodes in the Graph.
Once we find a minimum cut, we add an edge between the high degree
node and the target node of the local maximum flow computation to make
sure that we will not find that minimum cut again.
See also
--------
node_connectivity
edmonds_karp
shortest_augmenting_path
References
----------
.. [1] Kanevsky, A. (1993). Finding all minimum-size separating vertex
sets in a graph. Networks 23(6), 533--541.
http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract
"""
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is disconnected.')
# Initialize data structures.
# Keep track of the cuts already computed so we do not repeat them.
seen = []
# Even-Tarjan reduction is what we call auxiliary digraph
# for node connectivity.
H = build_auxiliary_node_connectivity(G)
mapping = H.graph['mapping']
R = build_residual_network(H, 'capacity')
kwargs = dict(capacity='capacity', residual=R)
# Define default flow function
if flow_func is None:
flow_func = default_flow_func
if flow_func is shortest_augmenting_path:
kwargs['two_phase'] = True
# Begin the actual algorithm
# step 1: Find node connectivity k of G
if k is None:
k = nx.node_connectivity(G, flow_func=flow_func)
# step 2:
# Find k nodes with top degree, call it X:
X = {n for n, d in sorted(G.degree(), key=itemgetter(1), reverse=True)[:k]}
# Check if X is a k-node-cutset
if _is_separating_set(G, X):
seen.append(X)
yield X
for x in X:
# step 3: Compute local connectivity flow of x with all other
# non adjacent nodes in G
non_adjacent = set(G) - X - set(G[x])
for v in non_adjacent:
# step 4: compute maximum flow in an Even-Tarjan reduction H of G
# and step:5 build the associated residual network R
R = flow_func(H, '%sB' % mapping[x], '%sA' % mapping[v], **kwargs)
flow_value = R.graph['flow_value']
if flow_value == k:
## Remove saturated edges form the residual network
saturated_edges = [(u, w, d)
for (u, w, d) in R.edges(data=True)
if d['capacity'] == d['flow']]
R.remove_edges_from(saturated_edges)
# step 6: shrink the strongly connected components of
# residual flow network R and call it L
L = nx.condensation(R)
cmap = L.graph['mapping']
# step 7: Compute antichains of L; they map to closed sets in H
# Any edge in H that links a closed set is part of a cutset
for antichain in nx.antichains(L):
# Nodes in an antichain of the condensation graph of
# the residual network map to a closed set of nodes that
# define a node partition of the auxiliary digraph H.
S = {n for n, scc in cmap.items() if scc in antichain}
# Find the cutset that links the node partition (S,~S) in H
cutset = set()
for u in S:
cutset.update((u, w) for w in H[u] if w not in S)
# The edges in H that form the cutset are internal edges
# (ie edges that represent a node of the original graph G)
node_cut = {
H.node[n]['id']
for edge in cutset for n in edge
}
if len(node_cut) == k:
if node_cut not in seen:
yield node_cut
seen.append(node_cut)
# Add an edge (x, v) to make sure that we do not
# find this cutset again. This is equivalent
# of adding the edge in the input graph
# G.add_edge(x, v) and then regenerate H and R:
# Add edges to the auxiliary digraph.
H.add_edge('%sB' % mapping[x],
'%sA' % mapping[v],
capacity=1)
H.add_edge('%sB' % mapping[v],
'%sA' % mapping[x],
capacity=1)
# Add edges to the residual network.
R.add_edge('%sB' % mapping[x],
'%sA' % mapping[v],
capacity=1)
R.add_edge('%sA' % mapping[v],
'%sB' % mapping[x],
capacity=1)
break
# Add again the saturated edges to reuse the residual network
R.add_edges_from(saturated_edges)
def minimum_edge_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of edges of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of edges of minimum cardinality that, if removed, would break
all paths among source and target in G. If not, it returns a set of
edges of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
cutset : set
Set of edges that, if removed, would disconnect G. If source
and target nodes are provided, the set contians the edges that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has edge connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_edge_cut(G))
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of minimum edge cut. For
undirected graphs the algorithm works by finding a 'small' dominating
set of nodes of G (see algorithm 7 in [1]_) and computing the maximum
flow between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_. For
directed graphs, the algorithm does n calls to the max flow function.
The function raises an error if the directed graph is not weakly
connected and returns an empty set if it is weakly connected.
It is an implementation of algorithm 8 in [1]_.
See also
--------
:meth:`minimum_st_edge_cut`
:meth:`minimum_node_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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.')
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)
# Local minimum edge cut if s and t are not None
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 minimum_st_edge_cut(H, s, t, **kwargs)
# Global minimum edge cut
# Analog to the algoritm for global edge connectivity
if G.is_directed():
# Based on algorithm 8 in [1]
if not nx.is_weakly_connected(G):
raise nx.NetworkXError('Input graph is not connected')
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
nodes = list(G)
n = len(nodes)
for i in range(n):
try:
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1],
**kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
except IndexError: # Last node!
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
else: # undirected
# Based on algorithm 6 in [1]
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is not connected')
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating set will always be of one node
# thus we return min_cut, which now contains the edges of a node
# with minimum degree
return min_cut
for w in D:
this_cut = minimum_st_edge_cut(H, v, w, **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
def edge_connectivity(G, s=None, t=None, flow_func=None):
r"""Returns the edge connectivity of the graph or digraph G.
The edge connectivity is equal to the minimum number of edges that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local edge
connectivity: the minimum number of edges that must be removed to
break all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected or directed graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
K : integer
Edge connectivity for G, or local edge connectivity if source
and target were provided
Examples
--------
>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(G)
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of global edge connectivity.
For undirected graphs the algorithm works by finding a 'small'
dominating set of nodes of G (see algorithm 7 in [1]_ ) and
computing local maximum flow (see :meth:`local_edge_connectivity`)
between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_ .
For directed graphs, the algorithm does n calls to the maximum
flow function. This is an implementation of algorithm 8 in [1]_ .
See also
--------
:meth:`local_edge_connectivity`
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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 edge 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_edge_connectivity(G, s, t, flow_func=flow_func)
# Global edge connectivity
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
if G.is_directed():
# Algorithm 8 in [1]
if not nx.is_weakly_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(G.degree().values())
nodes = G.nodes()
n = len(nodes)
for i in range(n):
kwargs['cutoff'] = L
try:
L = min(L, local_edge_connectivity(G, nodes[i], nodes[i+1],
**kwargs))
except IndexError: # last node!
L = min(L, local_edge_connectivity(G, nodes[i], nodes[0],
**kwargs))
return L
else: # undirected
# Algorithm 6 in [1]
if not nx.is_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(G.degree().values())
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating sets will always be of one node
# thus we return min degree
return L
for w in D:
kwargs['cutoff'] = L
L = min(L, local_edge_connectivity(G, v, w, **kwargs))
return L
def node_connectivity(G, s=None, t=None, flow_func=None):
"""Returns 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. 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.
Parameters
----------
G : NetworkX graph
Undirected graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
K : integer
Node connectivity of G, or local node connectivity if source
and target are provided.
Examples
--------
>>> # Platonic icosahedral graph is 5-node-connected
>>> G = nx.icosahedral_graph()
>>> nx.node_connectivity(G)
5
You can use alternative flow algorithms for the underlying maximum
flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local node connectivity.
>>> nx.node_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_node_connectivity` for details.
Notes
-----
This is a flow based implementation of node connectivity. The
algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2))`
maximum flow problems on an auxiliary digraph. Where `\delta`
is the minimum degree of G. For details about the auxiliary
digraph and the computation of local node connectivity see
:meth:`local_node_connectivity`. This implementation is based
on algorithm 11 in [1]_.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.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, flow_func=flow_func)
# Global node connectivity
if G.is_directed():
if not nx.is_weakly_connected(G):
return 0
iter_func = itertools.permutations
# It is necessary to consider both predecessors
# and successors for directed graphs
def neighbors(v):
return itertools.chain.from_iterable(
[G.predecessors(v), G.successors(v)])
else:
if not nx.is_connected(G):
return 0
iter_func = itertools.combinations
neighbors = G.neighbors
# Reuse the auxiliary digraph and the residual network
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
# Pick a node with minimum degree
# Node connectivity is bounded by degree.
v, K = min(G.degree(), key=itemgetter(1))
# compute local node connectivity with all its non-neighbors nodes
for w in set(G) - set(neighbors(v)) - set([v]):
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, v, w, **kwargs))
# Also for non adjacent pairs of neighbors of v
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, x, y, **kwargs))
return K
