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<h1>Source code for networkx.algorithms.connectivity.connectivity</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Flow based connectivity algorithms</span>
<span class="sd">"""</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">itemgetter</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="c1"># Define the default maximum flow function to use in all flow based</span>
<span class="c1"># connectivity algorithms.</span>
<span class="kn">from</span> <span class="nn">networkx.algorithms.flow</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">boykov_kolmogorov</span><span class="p">,</span>
<span class="n">build_residual_network</span><span class="p">,</span>
<span class="n">dinitz</span><span class="p">,</span>
<span class="n">edmonds_karp</span><span class="p">,</span>
<span class="n">shortest_augmenting_path</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">default_flow_func</span> <span class="o">=</span> <span class="n">edmonds_karp</span>
<span class="kn">from</span> <span class="nn">.utils</span> <span class="kn">import</span> <span class="n">build_auxiliary_edge_connectivity</span><span class="p">,</span> <span class="n">build_auxiliary_node_connectivity</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"average_node_connectivity"</span><span class="p">,</span>
<span class="s2">"local_node_connectivity"</span><span class="p">,</span>
<span class="s2">"node_connectivity"</span><span class="p">,</span>
<span class="s2">"local_edge_connectivity"</span><span class="p">,</span>
<span class="s2">"edge_connectivity"</span><span class="p">,</span>
<span class="s2">"all_pairs_node_connectivity"</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="local_node_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.local_node_connectivity.html#networkx.algorithms.connectivity.connectivity.local_node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">local_node_connectivity</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">auxiliary</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">residual</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Computes local node connectivity for nodes s and t.</span>
<span class="sd"> Local node connectivity for two non adjacent nodes s and t is the</span>
<span class="sd"> minimum number of nodes that must be removed (along with their incident</span>
<span class="sd"> edges) to disconnect them.</span>
<span class="sd"> This is a flow based implementation of node connectivity. We compute the</span>
<span class="sd"> maximum flow on an auxiliary digraph build from the original input</span>
<span class="sd"> graph (see below for details).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See below for details. The choice</span>
<span class="sd"> of the default function may change from version to version and</span>
<span class="sd"> should not be relied on. Default value: None.</span>
<span class="sd"> auxiliary : NetworkX DiGraph</span>
<span class="sd"> Auxiliary digraph to compute flow based node connectivity. It has</span>
<span class="sd"> to have a graph attribute called mapping with a dictionary mapping</span>
<span class="sd"> node names in G and in the auxiliary digraph. If provided</span>
<span class="sd"> it will be reused instead of recreated. Default value: None.</span>
<span class="sd"> residual : NetworkX DiGraph</span>
<span class="sd"> Residual network to compute maximum flow. If provided it will be</span>
<span class="sd"> reused instead of recreated. Default value: None.</span>
<span class="sd"> cutoff : integer, float, or None (default: None)</span>
<span class="sd"> If specified, the maximum flow algorithm will terminate when the</span>
<span class="sd"> flow value reaches or exceeds the cutoff. This only works for flows</span>
<span class="sd"> that support the cutoff parameter (most do) and is ignored otherwise.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> local node connectivity for nodes s and t</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> This function is not imported in the base NetworkX namespace, so you</span>
<span class="sd"> have to explicitly import it from the connectivity package:</span>
<span class="sd"> >>> from networkx.algorithms.connectivity import local_node_connectivity</span>
<span class="sd"> We use in this example the platonic icosahedral graph, which has node</span>
<span class="sd"> connectivity 5.</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> local_node_connectivity(G, 0, 6)</span>
<span class="sd"> 5</span>
<span class="sd"> If you need to compute local connectivity on several pairs of</span>
<span class="sd"> nodes in the same graph, it is recommended that you reuse the</span>
<span class="sd"> data structures that NetworkX uses in the computation: the</span>
<span class="sd"> auxiliary digraph for node connectivity, and the residual</span>
<span class="sd"> network for the underlying maximum flow computation.</span>
<span class="sd"> Example of how to compute local node connectivity among</span>
<span class="sd"> all pairs of nodes of the platonic icosahedral graph reusing</span>
<span class="sd"> the data structures.</span>
<span class="sd"> >>> import itertools</span>
<span class="sd"> >>> # You also have to explicitly import the function for</span>
<span class="sd"> >>> # building the auxiliary digraph from the connectivity package</span>
<span class="sd"> >>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity</span>
<span class="sd"> ...</span>
<span class="sd"> >>> H = build_auxiliary_node_connectivity(G)</span>
<span class="sd"> >>> # And the function for building the residual network from the</span>
<span class="sd"> >>> # flow package</span>
<span class="sd"> >>> from networkx.algorithms.flow import build_residual_network</span>
<span class="sd"> >>> # Note that the auxiliary digraph has an edge attribute named capacity</span>
<span class="sd"> >>> R = build_residual_network(H, "capacity")</span>
<span class="sd"> >>> result = dict.fromkeys(G, dict())</span>
<span class="sd"> >>> # Reuse the auxiliary digraph and the residual network by passing them</span>
<span class="sd"> >>> # as parameters</span>
<span class="sd"> >>> for u, v in itertools.combinations(G, 2):</span>
<span class="sd"> ... k = local_node_connectivity(G, u, v, auxiliary=H, residual=R)</span>
<span class="sd"> ... result[u][v] = k</span>
<span class="sd"> ...</span>
<span class="sd"> >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))</span>
<span class="sd"> True</span>
<span class="sd"> You can also use alternative flow algorithms for computing node</span>
<span class="sd"> connectivity. For instance, in dense networks the algorithm</span>
<span class="sd"> :meth:`shortest_augmenting_path` will usually perform better than</span>
<span class="sd"> the default :meth:`edmonds_karp` which is faster for sparse</span>
<span class="sd"> networks with highly skewed degree distributions. Alternative flow</span>
<span class="sd"> functions have to be explicitly imported from the flow package.</span>
<span class="sd"> >>> from networkx.algorithms.flow import shortest_augmenting_path</span>
<span class="sd"> >>> local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of node connectivity. We compute the</span>
<span class="sd"> maximum flow using, by default, the :meth:`edmonds_karp` algorithm (see:</span>
<span class="sd"> :meth:`maximum_flow`) on an auxiliary digraph build from the original</span>
<span class="sd"> input graph:</span>
<span class="sd"> For an undirected graph G having `n` nodes and `m` edges we derive a</span>
<span class="sd"> directed graph H with `2n` nodes and `2m+n` arcs by replacing each</span>
<span class="sd"> original node `v` with two nodes `v_A`, `v_B` linked by an (internal)</span>
<span class="sd"> arc in H. Then for each edge (`u`, `v`) in G we add two arcs</span>
<span class="sd"> (`u_B`, `v_A`) and (`v_B`, `u_A`) in H. Finally we set the attribute</span>
<span class="sd"> capacity = 1 for each arc in H [1]_ .</span>
<span class="sd"> For a directed graph G having `n` nodes and `m` arcs we derive a</span>
<span class="sd"> directed graph H with `2n` nodes and `m+n` arcs by replacing each</span>
<span class="sd"> original node `v` with two nodes `v_A`, `v_B` linked by an (internal)</span>
<span class="sd"> arc (`v_A`, `v_B`) in H. Then for each arc (`u`, `v`) in G we add one arc</span>
<span class="sd"> (`u_B`, `v_A`) in H. Finally we set the attribute capacity = 1 for</span>
<span class="sd"> each arc in H.</span>
<span class="sd"> This is equal to the local node connectivity because the value of</span>
<span class="sd"> a maximum s-t-flow is equal to the capacity of a minimum s-t-cut.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`local_edge_connectivity`</span>
<span class="sd"> :meth:`node_connectivity`</span>
<span class="sd"> :meth:`minimum_node_cut`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and</span>
<span class="sd"> Erlebach, 'Network Analysis: Methodological Foundations', Lecture</span>
<span class="sd"> Notes in Computer Science, Volume 3418, Springer-Verlag, 2005.</span>
<span class="sd"> http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">flow_func</span> <span class="o">=</span> <span class="n">default_flow_func</span>
<span class="k">if</span> <span class="n">auxiliary</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">auxiliary</span>
<span class="n">mapping</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">graph</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s2">"mapping"</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
<span class="k">if</span> <span class="n">mapping</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Invalid auxiliary digraph."</span><span class="p">)</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">residual</span><span class="p">}</span>
<span class="k">if</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">shortest_augmenting_path</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"two_phase"</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">edmonds_karp</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">dinitz</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">boykov_kolmogorov</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">maximum_flow_value</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">mapping</span><span class="p">[</span><span class="n">s</span><span class="p">]</span><span class="si">}</span><span class="s2">B"</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">mapping</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="si">}</span><span class="s2">A"</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
<div class="viewcode-block" id="node_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.node_connectivity.html#networkx.algorithms.connectivity.connectivity.node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns node connectivity for a graph or digraph G.</span>
<span class="sd"> Node connectivity is equal to the minimum number of nodes that</span>
<span class="sd"> must be removed to disconnect G or render it trivial. If source</span>
<span class="sd"> and target nodes are provided, this function returns the local node</span>
<span class="sd"> connectivity: the minimum number of nodes that must be removed to break</span>
<span class="sd"> all paths from source to target in G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional. Default value: None.</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional. Default value: None.</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See below for details. The</span>
<span class="sd"> choice of the default function may change from version</span>
<span class="sd"> to version and should not be relied on. Default value: None.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> Node connectivity of G, or local node connectivity if source</span>
<span class="sd"> and target are provided.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph is 5-node-connected</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.node_connectivity(G)</span>
<span class="sd"> 5</span>
<span class="sd"> You can use alternative flow algorithms for the underlying maximum</span>
<span class="sd"> flow computation. In dense networks the algorithm</span>
<span class="sd"> :meth:`shortest_augmenting_path` will usually perform better</span>
<span class="sd"> than the default :meth:`edmonds_karp`, which is faster for</span>
<span class="sd"> sparse networks with highly skewed degree distributions. Alternative</span>
<span class="sd"> flow functions have to be explicitly imported from the flow package.</span>
<span class="sd"> >>> from networkx.algorithms.flow import shortest_augmenting_path</span>
<span class="sd"> >>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)</span>
<span class="sd"> 5</span>
<span class="sd"> If you specify a pair of nodes (source and target) as parameters,</span>
<span class="sd"> this function returns the value of local node connectivity.</span>
<span class="sd"> >>> nx.node_connectivity(G, 3, 7)</span>
<span class="sd"> 5</span>
<span class="sd"> If you need to perform several local computations among different</span>
<span class="sd"> pairs of nodes on the same graph, it is recommended that you reuse</span>
<span class="sd"> the data structures used in the maximum flow computations. See</span>
<span class="sd"> :meth:`local_node_connectivity` for details.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of node connectivity. The</span>
<span class="sd"> algorithm works by solving $O((n-\delta-1+\delta(\delta-1)/2))$</span>
<span class="sd"> maximum flow problems on an auxiliary digraph. Where $\delta$</span>
<span class="sd"> is the minimum degree of G. For details about the auxiliary</span>
<span class="sd"> digraph and the computation of local node connectivity see</span>
<span class="sd"> :meth:`local_node_connectivity`. This implementation is based</span>
<span class="sd"> on algorithm 11 in [1]_.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`local_node_connectivity`</span>
<span class="sd"> :meth:`edge_connectivity`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="p">(</span><span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">s</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Both source and target must be specified."</span><span class="p">)</span>
<span class="c1"># Local node connectivity</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"node </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2"> not in graph"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"node </span><span class="si">{</span><span class="n">t</span><span class="si">}</span><span class="s2"> not in graph"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="n">flow_func</span><span class="p">)</span>
<span class="c1"># Global node connectivity</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="c1"># It is necessary to consider both predecessors</span>
<span class="c1"># and successors for directed graphs</span>
<span class="k">def</span> <span class="nf">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
<span class="k">return</span> <span class="n">itertools</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">([</span><span class="n">G</span><span class="o">.</span><span class="n">predecessors</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">G</span><span class="o">.</span><span class="n">successors</span><span class="p">(</span><span class="n">v</span><span class="p">)])</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">neighbors</span>
<span class="c1"># Reuse the auxiliary digraph and the residual network</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">build_residual_network</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="s2">"capacity"</span><span class="p">)</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"auxiliary"</span><span class="p">:</span> <span class="n">H</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">R</span><span class="p">}</span>
<span class="c1"># Pick a node with minimum degree</span>
<span class="c1"># Node connectivity is bounded by degree.</span>
<span class="n">v</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">(),</span> <span class="n">key</span><span class="o">=</span><span class="n">itemgetter</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
<span class="c1"># compute local node connectivity with all its non-neighbors nodes</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">-</span> <span class="p">{</span><span class="n">v</span><span class="p">}:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span>
<span class="n">K</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
<span class="c1"># Also for non adjacent pairs of neighbors of v</span>
<span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
<span class="k">if</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">x</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span>
<span class="n">K</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
<span class="k">return</span> <span class="n">K</span></div>
<div class="viewcode-block" id="average_node_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.average_node_connectivity.html#networkx.algorithms.connectivity.connectivity.average_node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">average_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns the average connectivity of a graph G.</span>
<span class="sd"> The average connectivity `\bar{\kappa}` of a graph G is the average</span>
<span class="sd"> of local node connectivity over all pairs of nodes of G [1]_ .</span>
<span class="sd"> .. math::</span>
<span class="sd"> \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity`</span>
<span class="sd"> for details. The choice of the default function may change from</span>
<span class="sd"> version to version and should not be relied on. Default value: None.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : float</span>
<span class="sd"> Average node connectivity</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`local_node_connectivity`</span>
<span class="sd"> :meth:`node_connectivity`</span>
<span class="sd"> :meth:`edge_connectivity`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average</span>
<span class="sd"> connectivity of a graph. Discrete mathematics 252(1-3), 31-45.</span>
<span class="sd"> http://www.sciencedirect.com/science/article/pii/S0012365X01001807</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="c1"># Reuse the auxiliary digraph and the residual network</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">build_residual_network</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="s2">"capacity"</span><span class="p">)</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"auxiliary"</span><span class="p">:</span> <span class="n">H</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">R</span><span class="p">}</span>
<span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">num</span> <span class="o">+=</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="n">den</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">den</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># Null Graph</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">num</span> <span class="o">/</span> <span class="n">den</span></div>
<div class="viewcode-block" id="all_pairs_node_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity.html#networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nbunch</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Compute node connectivity between all pairs of nodes of G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> nbunch: container</span>
<span class="sd"> Container of nodes. If provided node connectivity will be computed</span>
<span class="sd"> only over pairs of nodes in nbunch.</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See below for details. The</span>
<span class="sd"> choice of the default function may change from version</span>
<span class="sd"> to version and should not be relied on. Default value: None.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> all_pairs : dict</span>
<span class="sd"> A dictionary with node connectivity between all pairs of nodes</span>
<span class="sd"> in G, or in nbunch if provided.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`local_node_connectivity`</span>
<span class="sd"> :meth:`edge_connectivity`</span>
<span class="sd"> :meth:`local_edge_connectivity`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">nbunch</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">nbunch</span> <span class="o">=</span> <span class="n">G</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">nbunch</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">nbunch</span><span class="p">)</span>
<span class="n">directed</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">()</span>
<span class="k">if</span> <span class="n">directed</span><span class="p">:</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="n">all_pairs</span> <span class="o">=</span> <span class="p">{</span><span class="n">n</span><span class="p">:</span> <span class="p">{}</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nbunch</span><span class="p">}</span>
<span class="c1"># Reuse auxiliary digraph and residual network</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">mapping</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">graph</span><span class="p">[</span><span class="s2">"mapping"</span><span class="p">]</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">build_residual_network</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="s2">"capacity"</span><span class="p">)</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"auxiliary"</span><span class="p">:</span> <span class="n">H</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">R</span><span class="p">}</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">nbunch</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="n">all_pairs</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">directed</span><span class="p">:</span>
<span class="n">all_pairs</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span>
<span class="k">return</span> <span class="n">all_pairs</span></div>
<div class="viewcode-block" id="local_edge_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html#networkx.algorithms.connectivity.connectivity.local_edge_connectivity">[docs]</a><span class="k">def</span> <span class="nf">local_edge_connectivity</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">auxiliary</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">residual</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns local edge connectivity for nodes s and t in G.</span>
<span class="sd"> Local edge connectivity for two nodes s and t is the minimum number</span>
<span class="sd"> of edges that must be removed to disconnect them.</span>
<span class="sd"> This is a flow based implementation of edge connectivity. We compute the</span>
<span class="sd"> maximum flow on an auxiliary digraph build from the original</span>
<span class="sd"> network (see below for details). This is equal to the local edge</span>
<span class="sd"> connectivity because the value of a maximum s-t-flow is equal to the</span>
<span class="sd"> capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected or directed graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See below for details. The</span>
<span class="sd"> choice of the default function may change from version</span>
<span class="sd"> to version and should not be relied on. Default value: None.</span>
<span class="sd"> auxiliary : NetworkX DiGraph</span>
<span class="sd"> Auxiliary digraph for computing flow based edge connectivity. If</span>
<span class="sd"> provided it will be reused instead of recreated. Default value: None.</span>
<span class="sd"> residual : NetworkX DiGraph</span>
<span class="sd"> Residual network to compute maximum flow. If provided it will be</span>
<span class="sd"> reused instead of recreated. Default value: None.</span>
<span class="sd"> cutoff : integer, float, or None (default: None)</span>
<span class="sd"> If specified, the maximum flow algorithm will terminate when the</span>
<span class="sd"> flow value reaches or exceeds the cutoff. This only works for flows</span>
<span class="sd"> that support the cutoff parameter (most do) and is ignored otherwise.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> local edge connectivity for nodes s and t.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> This function is not imported in the base NetworkX namespace, so you</span>
<span class="sd"> have to explicitly import it from the connectivity package:</span>
<span class="sd"> >>> from networkx.algorithms.connectivity import local_edge_connectivity</span>
<span class="sd"> We use in this example the platonic icosahedral graph, which has edge</span>
<span class="sd"> connectivity 5.</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> local_edge_connectivity(G, 0, 6)</span>
<span class="sd"> 5</span>
<span class="sd"> If you need to compute local connectivity on several pairs of</span>
<span class="sd"> nodes in the same graph, it is recommended that you reuse the</span>
<span class="sd"> data structures that NetworkX uses in the computation: the</span>
<span class="sd"> auxiliary digraph for edge connectivity, and the residual</span>
<span class="sd"> network for the underlying maximum flow computation.</span>
<span class="sd"> Example of how to compute local edge connectivity among</span>
<span class="sd"> all pairs of nodes of the platonic icosahedral graph reusing</span>
<span class="sd"> the data structures.</span>
<span class="sd"> >>> import itertools</span>
<span class="sd"> >>> # You also have to explicitly import the function for</span>
<span class="sd"> >>> # building the auxiliary digraph from the connectivity package</span>
<span class="sd"> >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity</span>
<span class="sd"> >>> H = build_auxiliary_edge_connectivity(G)</span>
<span class="sd"> >>> # And the function for building the residual network from the</span>
<span class="sd"> >>> # flow package</span>
<span class="sd"> >>> from networkx.algorithms.flow import build_residual_network</span>
<span class="sd"> >>> # Note that the auxiliary digraph has an edge attribute named capacity</span>
<span class="sd"> >>> R = build_residual_network(H, "capacity")</span>
<span class="sd"> >>> result = dict.fromkeys(G, dict())</span>
<span class="sd"> >>> # Reuse the auxiliary digraph and the residual network by passing them</span>
<span class="sd"> >>> # as parameters</span>
<span class="sd"> >>> for u, v in itertools.combinations(G, 2):</span>
<span class="sd"> ... k = local_edge_connectivity(G, u, v, auxiliary=H, residual=R)</span>
<span class="sd"> ... result[u][v] = k</span>
<span class="sd"> >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))</span>
<span class="sd"> True</span>
<span class="sd"> You can also use alternative flow algorithms for computing edge</span>
<span class="sd"> connectivity. For instance, in dense networks the algorithm</span>
<span class="sd"> :meth:`shortest_augmenting_path` will usually perform better than</span>
<span class="sd"> the default :meth:`edmonds_karp` which is faster for sparse</span>
<span class="sd"> networks with highly skewed degree distributions. Alternative flow</span>
<span class="sd"> functions have to be explicitly imported from the flow package.</span>
<span class="sd"> >>> from networkx.algorithms.flow import shortest_augmenting_path</span>
<span class="sd"> >>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of edge connectivity. We compute the</span>
<span class="sd"> maximum flow using, by default, the :meth:`edmonds_karp` algorithm on an</span>
<span class="sd"> auxiliary digraph build from the original input graph:</span>
<span class="sd"> If the input graph is undirected, we replace each edge (`u`,`v`) with</span>
<span class="sd"> two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute</span>
<span class="sd"> 'capacity' for each arc to 1. If the input graph is directed we simply</span>
<span class="sd"> add the 'capacity' attribute. This is an implementation of algorithm 1</span>
<span class="sd"> in [1]_.</span>
<span class="sd"> The maximum flow in the auxiliary network is equal to the local edge</span>
<span class="sd"> connectivity because the value of a maximum s-t-flow is equal to the</span>
<span class="sd"> capacity of a minimum s-t-cut (Ford and Fulkerson theorem).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`edge_connectivity`</span>
<span class="sd"> :meth:`local_node_connectivity`</span>
<span class="sd"> :meth:`node_connectivity`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">flow_func</span> <span class="o">=</span> <span class="n">default_flow_func</span>
<span class="k">if</span> <span class="n">auxiliary</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">auxiliary</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">residual</span><span class="p">}</span>
<span class="k">if</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">shortest_augmenting_path</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"two_phase"</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">edmonds_karp</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">dinitz</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">elif</span> <span class="n">flow_func</span> <span class="ow">is</span> <span class="n">boykov_kolmogorov</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">cutoff</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">maximum_flow_value</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
<div class="viewcode-block" id="edge_connectivity"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.connectivity.connectivity.edge_connectivity.html#networkx.algorithms.connectivity.connectivity.edge_connectivity">[docs]</a><span class="k">def</span> <span class="nf">edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns the edge connectivity of the graph or digraph G.</span>
<span class="sd"> The edge connectivity is equal to the minimum number of edges that</span>
<span class="sd"> must be removed to disconnect G or render it trivial. If source</span>
<span class="sd"> and target nodes are provided, this function returns the local edge</span>
<span class="sd"> connectivity: the minimum number of edges that must be removed to</span>
<span class="sd"> break all paths from source to target in G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected or directed graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional. Default value: None.</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional. Default value: None.</span>
<span class="sd"> flow_func : function</span>
<span class="sd"> A function for computing the maximum flow among a pair of nodes.</span>
<span class="sd"> The function has to accept at least three parameters: a Digraph,</span>
<span class="sd"> a source node, and a target node. And return a residual network</span>
<span class="sd"> that follows NetworkX conventions (see :meth:`maximum_flow` for</span>
<span class="sd"> details). If flow_func is None, the default maximum flow function</span>
<span class="sd"> (:meth:`edmonds_karp`) is used. See below for details. The</span>
<span class="sd"> choice of the default function may change from version</span>
<span class="sd"> to version and should not be relied on. Default value: None.</span>
<span class="sd"> cutoff : integer, float, or None (default: None)</span>
<span class="sd"> If specified, the maximum flow algorithm will terminate when the</span>
<span class="sd"> flow value reaches or exceeds the cutoff. This only works for flows</span>
<span class="sd"> that support the cutoff parameter (most do) and is ignored otherwise.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> Edge connectivity for G, or local edge connectivity if source</span>
<span class="sd"> and target were provided</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph is 5-edge-connected</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.edge_connectivity(G)</span>
<span class="sd"> 5</span>
<span class="sd"> You can use alternative flow algorithms for the underlying</span>
<span class="sd"> maximum flow computation. In dense networks the algorithm</span>
<span class="sd"> :meth:`shortest_augmenting_path` will usually perform better</span>
<span class="sd"> than the default :meth:`edmonds_karp`, which is faster for</span>
<span class="sd"> sparse networks with highly skewed degree distributions.</span>
<span class="sd"> Alternative flow functions have to be explicitly imported</span>
<span class="sd"> from the flow package.</span>
<span class="sd"> >>> from networkx.algorithms.flow import shortest_augmenting_path</span>
<span class="sd"> >>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)</span>
<span class="sd"> 5</span>
<span class="sd"> If you specify a pair of nodes (source and target) as parameters,</span>
<span class="sd"> this function returns the value of local edge connectivity.</span>
<span class="sd"> >>> nx.edge_connectivity(G, 3, 7)</span>
<span class="sd"> 5</span>
<span class="sd"> If you need to perform several local computations among different</span>
<span class="sd"> pairs of nodes on the same graph, it is recommended that you reuse</span>
<span class="sd"> the data structures used in the maximum flow computations. See</span>
<span class="sd"> :meth:`local_edge_connectivity` for details.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of global edge connectivity.</span>
<span class="sd"> For undirected graphs the algorithm works by finding a 'small'</span>
<span class="sd"> dominating set of nodes of G (see algorithm 7 in [1]_ ) and</span>
<span class="sd"> computing local maximum flow (see :meth:`local_edge_connectivity`)</span>
<span class="sd"> between an arbitrary node in the dominating set and the rest of</span>
<span class="sd"> nodes in it. This is an implementation of algorithm 6 in [1]_ .</span>
<span class="sd"> For directed graphs, the algorithm does n calls to the maximum</span>
<span class="sd"> flow function. This is an implementation of algorithm 8 in [1]_ .</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> :meth:`local_edge_connectivity`</span>
<span class="sd"> :meth:`local_node_connectivity`</span>
<span class="sd"> :meth:`node_connectivity`</span>
<span class="sd"> :meth:`maximum_flow`</span>
<span class="sd"> :meth:`edmonds_karp`</span>
<span class="sd"> :meth:`preflow_push`</span>
<span class="sd"> :meth:`shortest_augmenting_path`</span>
<span class="sd"> :meth:`k_edge_components`</span>
<span class="sd"> :meth:`k_edge_subgraphs`</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="p">(</span><span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">s</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Both source and target must be specified."</span><span class="p">)</span>
<span class="c1"># Local edge connectivity</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"node </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2"> not in graph"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"node </span><span class="si">{</span><span class="n">t</span><span class="si">}</span><span class="s2"> not in graph"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">flow_func</span><span class="o">=</span><span class="n">flow_func</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">)</span>
<span class="c1"># Global edge connectivity</span>
<span class="c1"># reuse auxiliary digraph and residual network</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">build_auxiliary_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">build_residual_network</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="s2">"capacity"</span><span class="p">)</span>
<span class="n">kwargs</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"flow_func"</span><span class="p">:</span> <span class="n">flow_func</span><span class="p">,</span> <span class="s2">"auxiliary"</span><span class="p">:</span> <span class="n">H</span><span class="p">,</span> <span class="s2">"residual"</span><span class="p">:</span> <span class="n">R</span><span class="p">}</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="c1"># Algorithm 8 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="c1"># initial value for \lambda is minimum degree</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">d</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodes</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cutoff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">L</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span> <span class="c1"># last node!</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
<span class="k">return</span> <span class="n">L</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># undirected</span>
<span class="c1"># Algorithm 6 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="c1"># initial value for \lambda is minimum degree</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">d</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
<span class="k">if</span> <span class="n">cutoff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span>
<span class="c1"># A dominating set is \lambda-covering</span>
<span class="c1"># We need a dominating set with at least two nodes</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">dominating_set</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">start_with</span><span class="o">=</span><span class="n">node</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">D</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># in complete graphs the dominating sets will always be of one node</span>
<span class="c1"># thus we return min degree</span>
<span class="k">return</span> <span class="n">L</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">D</span><span class="p">:</span>
<span class="n">kwargs</span><span class="p">[</span><span class="s2">"cutoff"</span><span class="p">]</span> <span class="o">=</span> <span class="n">L</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
<span class="k">return</span> <span class="n">L</span></div>
</pre></div>
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