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This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.approximation.connectivity

""" Fast approximation for node connectivity
"""
#    Jordi Torrents <jtorrents@milnou.net>
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 icosahedral graph has node connectivity 5 >>> # for each non adjacent node pair >>> from networkx.algorithms import approximation as approx >>> G = nx.icosahedral_graph() >>> approx.local_node_connectivity(G, 0, 6) 5 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 icosahedral graph is 5-node-connected >>> from networkx.algorithms import approximation as approx >>> G = nx.icosahedral_graph() >>> approx.node_connectivity(G) 5 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.from_iterable([G.predecessors_iter(v), G.successors_iter(v)]) else: connected_func = nx.is_connected iter_func = itertools.combinations neighbors = G.neighbors_iter if not connected_func(G): return 0 # Choose a node with minimum degree v, minimum_degree = min(G.degree().items(), 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 originaly 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_iter Gsucc = G.successors_iter else: Gpred = G.neighbors_iter Gsucc = G.neighbors_iter # 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 tigger "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))