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## iof-tools / networkxMiCe / networkx-master / networkx / algorithms / connectivity / connectivity.py @ 5cef0f13

 1 # -*- coding: utf-8 -*-  """  Flow based connectivity algorithms  """  from __future__ import division  import itertools  from operator import itemgetter  import networkx as nx  # Define the default maximum flow function to use in all flow based  # connectivity algorithms.  from networkx.algorithms.flow import boykov_kolmogorov  from networkx.algorithms.flow import dinitz  from networkx.algorithms.flow import edmonds_karp  from networkx.algorithms.flow import shortest_augmenting_path  from networkx.algorithms.flow import build_residual_network  default_flow_func = edmonds_karp  from .utils import (build_auxiliary_node_connectivity,   build_auxiliary_edge_connectivity)  __author__ = '\n'.join(['Jordi Torrents '])  __all__ = ['average_node_connectivity',   'local_node_connectivity',   'node_connectivity',   'local_edge_connectivity',   'edge_connectivity',   'all_pairs_node_connectivity']  def local_node_connectivity(G, s, t, flow_func=None, auxiliary=None,   residual=None, cutoff=None):   r"""Computes local node connectivity for nodes s and t.     Local node connectivity for two non adjacent nodes s and t is the   minimum number of nodes that must be removed (along with their incident   edges) to disconnect them.     This is a flow based implementation of node connectivity. We compute the   maximum flow on an auxiliary digraph build from the original input   graph (see below for details).     Parameters   ----------   G : NetworkX graph   Undirected graph     s : node   Source node     t : node   Target node     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.     auxiliary : NetworkX DiGraph   Auxiliary digraph to compute flow based node connectivity. It has   to have a graph attribute called mapping with a dictionary mapping   node names in G and in the auxiliary digraph. If provided   it will be reused instead of recreated. Default value: None.     residual : NetworkX DiGraph   Residual network to compute maximum flow. If provided it will be   reused instead of recreated. Default value: None.     cutoff : integer, float   If specified, the maximum flow algorithm will terminate when the   flow value reaches or exceeds the cutoff. This is only for the   algorithms that support the cutoff parameter: :meth:edmonds_karp   and :meth:shortest_augmenting_path. Other algorithms will ignore   this parameter. Default value: None.     Returns   -------   K : integer   local node connectivity for nodes s and t     Examples   --------   This function is not imported in the base NetworkX namespace, so you   have to explicitly import it from the connectivity package:     >>> from networkx.algorithms.connectivity import local_node_connectivity     We use in this example the platonic icosahedral graph, which has node   connectivity 5.     >>> G = nx.icosahedral_graph()   >>> local_node_connectivity(G, 0, 6)   5     If you need to compute local connectivity on several pairs of   nodes in the same graph, it is recommended that you reuse the   data structures that NetworkX uses in the computation: the   auxiliary digraph for node connectivity, and the residual   network for the underlying maximum flow computation.     Example of how to compute local node connectivity among   all pairs of nodes of the platonic icosahedral graph reusing   the data structures.     >>> import itertools   >>> # You also have to explicitly import the function for   >>> # building the auxiliary digraph from the connectivity package   >>> from networkx.algorithms.connectivity import (   ... build_auxiliary_node_connectivity)   ...   >>> H = build_auxiliary_node_connectivity(G)   >>> # And the function for building the residual network from the   >>> # flow package   >>> from networkx.algorithms.flow import build_residual_network   >>> # Note that the auxiliary digraph has an edge attribute named capacity   >>> R = build_residual_network(H, 'capacity')   >>> result = dict.fromkeys(G, dict())   >>> # Reuse the auxiliary digraph and the residual network by passing them   >>> # as parameters   >>> for u, v in itertools.combinations(G, 2):   ... k = local_node_connectivity(G, u, v, auxiliary=H, residual=R)   ... result[u][v] = k   ...   >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))   True     You can also use alternative flow algorithms for computing node   connectivity. For instance, 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   >>> local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)   5     Notes   -----   This is a flow based implementation of node connectivity. We compute the   maximum flow using, by default, the :meth:edmonds_karp algorithm (see:   :meth:maximum_flow) on an auxiliary digraph build from the original   input graph:     For an undirected graph G having n nodes and m edges we derive a   directed graph H with 2n nodes and 2m+n arcs by replacing each   original node v with two nodes v_A, v_B linked by an (internal)   arc in H. Then for each edge (u, v) in G we add two arcs   (u_B, v_A) and (v_B, u_A) in H. Finally we set the attribute   capacity = 1 for each arc in H [1]_ .     For a directed graph G having n nodes and m arcs we derive a   directed graph H with 2n nodes and m+n arcs by replacing each   original node v with two nodes v_A, v_B linked by an (internal)   arc (v_A, v_B) in H. Then for each arc (u, v) in G we add one arc   (u_B, v_A) in H. Finally we set the attribute capacity = 1 for   each arc in H.     This is equal to the local node connectivity because the value of   a maximum s-t-flow is equal to the capacity of a minimum s-t-cut.     See also   --------   :meth:local_edge_connectivity   :meth:node_connectivity   :meth:minimum_node_cut   :meth:maximum_flow   :meth:edmonds_karp   :meth:preflow_push   :meth:shortest_augmenting_path     References   ----------   .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and   Erlebach, 'Network Analysis: Methodological Foundations', Lecture   Notes in Computer Science, Volume 3418, Springer-Verlag, 2005.   http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf     """   if flow_func is None:   flow_func = default_flow_func   if auxiliary is None:   H = build_auxiliary_node_connectivity(G)   else:   H = auxiliary   mapping = H.graph.get('mapping', None)   if mapping is None:   raise nx.NetworkXError('Invalid auxiliary digraph.')   kwargs = dict(flow_func=flow_func, residual=residual)   if flow_func is shortest_augmenting_path:   kwargs['cutoff'] = cutoff   kwargs['two_phase'] = True   elif flow_func is edmonds_karp:   kwargs['cutoff'] = cutoff   elif flow_func is dinitz:   kwargs['cutoff'] = cutoff   elif flow_func is boykov_kolmogorov:   kwargs['cutoff'] = cutoff   return nx.maximum_flow_value(H, '%sB' % mapping[s], '%sA' % mapping[t], **kwargs)  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(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  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 local_edge_connectivity(G, s, t, flow_func=None, auxiliary=None,   residual=None, cutoff=None):   r"""Returns local edge connectivity for nodes s and t in G.     Local edge connectivity for two nodes s and t is the minimum number   of edges that must be removed to disconnect them.     This is a flow based implementation of edge connectivity. We compute the   maximum flow on an auxiliary digraph build from the original   network (see below for details). This is equal to the local edge   connectivity because the value of a maximum s-t-flow is equal to the   capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .     Parameters   ----------   G : NetworkX graph   Undirected or directed graph     s : node   Source node     t : node   Target node     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.     auxiliary : NetworkX DiGraph   Auxiliary digraph for computing flow based edge connectivity. If   provided it will be reused instead of recreated. Default value: None.     residual : NetworkX DiGraph   Residual network to compute maximum flow. If provided it will be   reused instead of recreated. Default value: None.     cutoff : integer, float   If specified, the maximum flow algorithm will terminate when the   flow value reaches or exceeds the cutoff. This is only for the   algorithms that support the cutoff parameter: :meth:edmonds_karp   and :meth:shortest_augmenting_path. Other algorithms will ignore   this parameter. Default value: None.     Returns   -------   K : integer   local edge connectivity for nodes s and t.     Examples   --------   This function is not imported in the base NetworkX namespace, so you   have to explicitly import it from the connectivity package:     >>> from networkx.algorithms.connectivity import local_edge_connectivity     We use in this example the platonic icosahedral graph, which has edge   connectivity 5.     >>> G = nx.icosahedral_graph()   >>> local_edge_connectivity(G, 0, 6)   5     If you need to compute local connectivity on several pairs of   nodes in the same graph, it is recommended that you reuse the   data structures that NetworkX uses in the computation: the   auxiliary digraph for edge connectivity, and the residual   network for the underlying maximum flow computation.     Example of how to compute local edge connectivity among   all pairs of nodes of the platonic icosahedral graph reusing   the data structures.     >>> import itertools   >>> # You also have to explicitly import the function for   >>> # building the auxiliary digraph from the connectivity package   >>> from networkx.algorithms.connectivity import (   ... build_auxiliary_edge_connectivity)   >>> H = build_auxiliary_edge_connectivity(G)   >>> # And the function for building the residual network from the   >>> # flow package   >>> from networkx.algorithms.flow import build_residual_network   >>> # Note that the auxiliary digraph has an edge attribute named capacity   >>> R = build_residual_network(H, 'capacity')   >>> result = dict.fromkeys(G, dict())   >>> # Reuse the auxiliary digraph and the residual network by passing them   >>> # as parameters   >>> for u, v in itertools.combinations(G, 2):   ... k = local_edge_connectivity(G, u, v, auxiliary=H, residual=R)   ... result[u][v] = k   >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))   True     You can also use alternative flow algorithms for computing edge   connectivity. For instance, 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   >>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)   5     Notes   -----   This is a flow based implementation of edge connectivity. We compute the   maximum flow using, by default, the :meth:edmonds_karp algorithm on an   auxiliary digraph build from the original input graph:     If the input graph is undirected, we replace each edge (u,v) with   two reciprocal arcs (u, v) and (v, u) and then we set the attribute   'capacity' for each arc to 1. If the input graph is directed we simply   add the 'capacity' attribute. This is an implementation of algorithm 1   in [1]_.     The maximum flow in the auxiliary network is equal to the local edge   connectivity because the value of a maximum s-t-flow is equal to the   capacity of a minimum s-t-cut (Ford and Fulkerson theorem).     See also   --------   :meth: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 flow_func is None:   flow_func = default_flow_func   if auxiliary is None:   H = build_auxiliary_edge_connectivity(G)   else:   H = auxiliary   kwargs = dict(flow_func=flow_func, residual=residual)   if flow_func is shortest_augmenting_path:   kwargs['cutoff'] = cutoff   kwargs['two_phase'] = True   elif flow_func is edmonds_karp:   kwargs['cutoff'] = cutoff   elif flow_func is dinitz:   kwargs['cutoff'] = cutoff   elif flow_func is boykov_kolmogorov:   kwargs['cutoff'] = cutoff   return nx.maximum_flow_value(H, s, t, **kwargs)  def edge_connectivity(G, s=None, t=None, flow_func=None, cutoff=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.     cutoff : integer, float   If specified, the maximum flow algorithm will terminate when the   flow value reaches or exceeds the cutoff. This is only for the   algorithms that support the cutoff parameter: :meth:edmonds_karp   and :meth:shortest_augmenting_path. Other algorithms will ignore   this parameter. 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,   cutoff=cutoff)   # 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)   if cutoff is not None:   L = min(cutoff, L)   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())   if cutoff is not None:   L = min(cutoff, L)   # 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