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""" Flow based connectivity algorithms """ 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 __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, f"{mapping[s]}B", f"{mapping[t]}A", **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(f"node {s} not in graph") if t not in G: raise nx.NetworkXError(f"node {t} not in graph") 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)) - {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: e.g., :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(f"node {s} not in graph") if t not in G: raise nx.NetworkXError(f"node {t} not in graph") 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