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<h1>Source code for networkx.algorithms.threshold</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Threshold Graphs - Creation, manipulation and identification.</span>
<span class="sd">"""</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">py_random_state</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"is_threshold_graph"</span><span class="p">,</span> <span class="s2">"find_threshold_graph"</span><span class="p">]</span>
<div class="viewcode-block" id="is_threshold_graph"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.threshold.is_threshold_graph.html#networkx.algorithms.threshold.is_threshold_graph">[docs]</a><span class="k">def</span> <span class="nf">is_threshold_graph</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns `True` if `G` is a threshold graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph instance</span>
<span class="sd"> An instance of `Graph`, `DiGraph`, `MultiGraph` or `MultiDiGraph`</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> `True` if `G` is a threshold graph, `False` otherwise.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms.threshold import is_threshold_graph</span>
<span class="sd"> >>> G = nx.path_graph(3)</span>
<span class="sd"> >>> is_threshold_graph(G)</span>
<span class="sd"> True</span>
<span class="sd"> >>> G = nx.barbell_graph(3, 3)</span>
<span class="sd"> >>> is_threshold_graph(G)</span>
<span class="sd"> False</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">is_threshold_sequence</span><span class="p">(</span><span class="nb">list</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></div>
<span class="k">def</span> <span class="nf">is_threshold_sequence</span><span class="p">(</span><span class="n">degree_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns True if the sequence is a threshold degree seqeunce.</span>
<span class="sd"> Uses the property that a threshold graph must be constructed by</span>
<span class="sd"> adding either dominating or isolated nodes. Thus, it can be</span>
<span class="sd"> deconstructed iteratively by removing a node of degree zero or a</span>
<span class="sd"> node that connects to the remaining nodes. If this deconstruction</span>
<span class="sd"> failes then the sequence is not a threshold sequence.</span>
<span class="sd"> """</span>
<span class="n">ds</span> <span class="o">=</span> <span class="n">degree_sequence</span><span class="p">[:]</span> <span class="c1"># get a copy so we don't destroy original</span>
<span class="n">ds</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
<span class="k">while</span> <span class="n">ds</span><span class="p">:</span>
<span class="k">if</span> <span class="n">ds</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># if isolated node</span>
<span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1"># remove it</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="n">ds</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># is the largest degree node dominating?</span>
<span class="k">return</span> <span class="kc">False</span> <span class="c1"># no, not a threshold degree sequence</span>
<span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span> <span class="c1"># yes, largest is the dominating node</span>
<span class="n">ds</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ds</span><span class="p">]</span> <span class="c1"># remove it and decrement all degrees</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">def</span> <span class="nf">creation_sequence</span><span class="p">(</span><span class="n">degree_sequence</span><span class="p">,</span> <span class="n">with_labels</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">compact</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Determines the creation sequence for the given threshold degree sequence.</span>
<span class="sd"> The creation sequence is a list of single characters 'd'</span>
<span class="sd"> or 'i': 'd' for dominating or 'i' for isolated vertices.</span>
<span class="sd"> Dominating vertices are connected to all vertices present when it</span>
<span class="sd"> is added. The first node added is by convention 'd'.</span>
<span class="sd"> This list can be converted to a string if desired using "".join(cs)</span>
<span class="sd"> If with_labels==True:</span>
<span class="sd"> Returns a list of 2-tuples containing the vertex number</span>
<span class="sd"> and a character 'd' or 'i' which describes the type of vertex.</span>
<span class="sd"> If compact==True:</span>
<span class="sd"> Returns the creation sequence in a compact form that is the number</span>
<span class="sd"> of 'i's and 'd's alternating.</span>
<span class="sd"> Examples:</span>
<span class="sd"> [1,2,2,3] represents d,i,i,d,d,i,i,i</span>
<span class="sd"> [3,1,2] represents d,d,d,i,d,d</span>
<span class="sd"> Notice that the first number is the first vertex to be used for</span>
<span class="sd"> construction and so is always 'd'.</span>
<span class="sd"> with_labels and compact cannot both be True.</span>
<span class="sd"> Returns None if the sequence is not a threshold sequence</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">with_labels</span> <span class="ow">and</span> <span class="n">compact</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"compact sequences cannot be labeled"</span><span class="p">)</span>
<span class="c1"># make an indexed copy</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">degree_sequence</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span> <span class="c1"># labeled degree seqeunce</span>
<span class="n">ds</span> <span class="o">=</span> <span class="p">[[</span><span class="n">degree</span><span class="p">,</span> <span class="n">label</span><span class="p">]</span> <span class="k">for</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span> <span class="ow">in</span> <span class="n">degree_sequence</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">ds</span> <span class="o">=</span> <span class="p">[[</span><span class="n">d</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">degree_sequence</span><span class="p">)]</span>
<span class="n">ds</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># creation sequence</span>
<span class="k">while</span> <span class="n">ds</span><span class="p">:</span>
<span class="k">if</span> <span class="n">ds</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># isolated node</span>
<span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">=</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># make sure we start with a d</span>
<span class="n">cs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="s2">"i"</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="s2">"d"</span><span class="p">))</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="n">ds</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># Not dominating node</span>
<span class="k">return</span> <span class="kc">None</span> <span class="c1"># not a threshold degree sequence</span>
<span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">=</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">cs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="s2">"d"</span><span class="p">))</span>
<span class="n">ds</span> <span class="o">=</span> <span class="p">[[</span><span class="n">d</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ds</span><span class="p">]</span> <span class="c1"># decrement due to removing node</span>
<span class="k">if</span> <span class="n">with_labels</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="k">if</span> <span class="n">compact</span><span class="p">:</span>
<span class="k">return</span> <span class="n">make_compact</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
<span class="k">return</span> <span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">cs</span><span class="p">]</span> <span class="c1"># not labeled</span>
<span class="k">def</span> <span class="nf">make_compact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns the creation sequence in a compact form</span>
<span class="sd"> that is the number of 'i's and 'd's alternating.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms.threshold import make_compact</span>
<span class="sd"> >>> make_compact(["d", "i", "i", "d", "d", "i", "i", "i"])</span>
<span class="sd"> [1, 2, 2, 3]</span>
<span class="sd"> >>> make_compact(["d", "d", "d", "i", "d", "d"])</span>
<span class="sd"> [3, 1, 2]</span>
<span class="sd"> Notice that the first number is the first vertex</span>
<span class="sd"> to be used for construction and so is always 'd'.</span>
<span class="sd"> Labeled creation sequences lose their labels in the</span>
<span class="sd"> compact representation.</span>
<span class="sd"> >>> make_compact([3, 1, 2])</span>
<span class="sd"> [3, 1, 2]</span>
<span class="sd"> """</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">creation_sequence</span><span class="p">]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="k">return</span> <span class="n">creation_sequence</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Not a valid creation sequence type"</span><span class="p">)</span>
<span class="n">ccs</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># count the run lengths of d's or i's.</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="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">cs</span><span class="p">)):</span>
<span class="k">if</span> <span class="n">cs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">cs</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="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">ccs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">count</span><span class="p">)</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">ccs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">count</span><span class="p">)</span> <span class="c1"># don't forget the last one</span>
<span class="k">return</span> <span class="n">ccs</span>
<span class="k">def</span> <span class="nf">uncompact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Converts a compact creation sequence for a threshold</span>
<span class="sd"> graph to a standard creation sequence (unlabeled).</span>
<span class="sd"> If the creation_sequence is already standard, return it.</span>
<span class="sd"> See creation_sequence.</span>
<span class="sd"> """</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="k">return</span> <span class="n">creation_sequence</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="k">return</span> <span class="n">creation_sequence</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="n">ccscopy</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Not a valid creation sequence type"</span><span class="p">)</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="n">ccscopy</span><span class="p">:</span>
<span class="n">cs</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">ccscopy</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="p">[</span><span class="s2">"d"</span><span class="p">])</span>
<span class="k">if</span> <span class="n">ccscopy</span><span class="p">:</span>
<span class="n">cs</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">ccscopy</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="p">[</span><span class="s2">"i"</span><span class="p">])</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="k">def</span> <span class="nf">creation_sequence_to_weights</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns a list of node weights which create the threshold</span>
<span class="sd"> graph designated by the creation sequence. The weights</span>
<span class="sd"> are scaled so that the threshold is 1.0. The order of the</span>
<span class="sd"> nodes is the same as that in the creation sequence.</span>
<span class="sd"> """</span>
<span class="c1"># Turn input sequence into a labeled creation sequence</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span> <span class="c1"># string like 'ddidid'</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">creation_sequence</span><span class="p">]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="n">uncompact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Not a valid creation sequence type"</span><span class="p">)</span>
<span class="c1"># pass through twice--first backwards</span>
<span class="n">wseq</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="n">w</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">prev</span> <span class="o">=</span> <span class="s2">"i"</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">wseq</span><span class="p">):</span>
<span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="s2">"i"</span><span class="p">:</span>
<span class="n">wseq</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">w</span>
<span class="n">prev</span> <span class="o">=</span> <span class="n">s</span>
<span class="k">elif</span> <span class="n">prev</span> <span class="o">==</span> <span class="s2">"i"</span><span class="p">:</span>
<span class="n">prev</span> <span class="o">=</span> <span class="n">s</span>
<span class="n">w</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">wseq</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span> <span class="c1"># now pass through forwards</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">wseq</span><span class="p">):</span>
<span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">wseq</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">w</span>
<span class="n">prev</span> <span class="o">=</span> <span class="n">s</span>
<span class="k">elif</span> <span class="n">prev</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">prev</span> <span class="o">=</span> <span class="n">s</span>
<span class="n">w</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="c1"># Now scale weights</span>
<span class="k">if</span> <span class="n">prev</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">w</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">wscale</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">w</span>
<span class="k">return</span> <span class="p">[</span><span class="n">ww</span> <span class="o">*</span> <span class="n">wscale</span> <span class="k">for</span> <span class="n">ww</span> <span class="ow">in</span> <span class="n">wseq</span><span class="p">]</span>
<span class="c1"># return wseq</span>
<span class="k">def</span> <span class="nf">weights_to_creation_sequence</span><span class="p">(</span>
<span class="n">weights</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">with_labels</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">compact</span><span class="o">=</span><span class="kc">False</span>
<span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns a creation sequence for a threshold graph</span>
<span class="sd"> determined by the weights and threshold given as input.</span>
<span class="sd"> If the sum of two node weights is greater than the</span>
<span class="sd"> threshold value, an edge is created between these nodes.</span>
<span class="sd"> The creation sequence is a list of single characters 'd'</span>
<span class="sd"> or 'i': 'd' for dominating or 'i' for isolated vertices.</span>
<span class="sd"> Dominating vertices are connected to all vertices present</span>
<span class="sd"> when it is added. The first node added is by convention 'd'.</span>
<span class="sd"> If with_labels==True:</span>
<span class="sd"> Returns a list of 2-tuples containing the vertex number</span>
<span class="sd"> and a character 'd' or 'i' which describes the type of vertex.</span>
<span class="sd"> If compact==True:</span>
<span class="sd"> Returns the creation sequence in a compact form that is the number</span>
<span class="sd"> of 'i's and 'd's alternating.</span>
<span class="sd"> Examples:</span>
<span class="sd"> [1,2,2,3] represents d,i,i,d,d,i,i,i</span>
<span class="sd"> [3,1,2] represents d,d,d,i,d,d</span>
<span class="sd"> Notice that the first number is the first vertex to be used for</span>
<span class="sd"> construction and so is always 'd'.</span>
<span class="sd"> with_labels and compact cannot both be True.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">with_labels</span> <span class="ow">and</span> <span class="n">compact</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"compact sequences cannot be labeled"</span><span class="p">)</span>
<span class="c1"># make an indexed copy</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span> <span class="c1"># labeled weights</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">w</span><span class="p">,</span> <span class="n">label</span><span class="p">]</span> <span class="k">for</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span> <span class="ow">in</span> <span class="n">weights</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">wseq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">w</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">weights</span><span class="p">)]</span>
<span class="n">wseq</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># creation sequence</span>
<span class="n">cutoff</span> <span class="o">=</span> <span class="n">threshold</span> <span class="o">-</span> <span class="n">wseq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="k">while</span> <span class="n">wseq</span><span class="p">:</span>
<span class="k">if</span> <span class="n">wseq</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o"><</span> <span class="n">cutoff</span><span class="p">:</span> <span class="c1"># isolated node</span>
<span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span> <span class="o">=</span> <span class="n">wseq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">label</span><span class="p">,</span> <span class="s2">"i"</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span> <span class="o">=</span> <span class="n">wseq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">label</span><span class="p">,</span> <span class="s2">"d"</span><span class="p">))</span>
<span class="n">cutoff</span> <span class="o">=</span> <span class="n">threshold</span> <span class="o">-</span> <span class="n">wseq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">wseq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># make sure we start with a d</span>
<span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span> <span class="o">=</span> <span class="n">wseq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">label</span><span class="p">,</span> <span class="s2">"d"</span><span class="p">))</span>
<span class="c1"># put in correct order</span>
<span class="n">cs</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="k">if</span> <span class="n">with_labels</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="k">if</span> <span class="n">compact</span><span class="p">:</span>
<span class="k">return</span> <span class="n">make_compact</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
<span class="k">return</span> <span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">cs</span><span class="p">]</span> <span class="c1"># not labeled</span>
<span class="c1"># Manipulating NetworkX.Graphs in context of threshold graphs</span>
<span class="k">def</span> <span class="nf">threshold_graph</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">,</span> <span class="n">create_using</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Create a threshold graph from the creation sequence or compact</span>
<span class="sd"> creation_sequence.</span>
<span class="sd"> The input sequence can be a</span>
<span class="sd"> creation sequence (e.g. ['d','i','d','d','d','i'])</span>
<span class="sd"> labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])</span>
<span class="sd"> compact creation sequence (e.g. [2,1,1,2,0])</span>
<span class="sd"> Use cs=creation_sequence(degree_sequence,labeled=True)</span>
<span class="sd"> to convert a degree sequence to a creation sequence.</span>
<span class="sd"> Returns None if the sequence is not valid</span>
<span class="sd"> """</span>
<span class="c1"># Turn input sequence into a labeled creation sequence</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="n">ci</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">enumerate</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">))</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="n">ci</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">uncompact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">ci</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"not a valid creation sequence type"</span><span class="p">)</span>
<span class="k">return</span> <span class="kc">None</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">empty_graph</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">create_using</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="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Directed Graph not supported"</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="s2">"Threshold Graph"</span>
<span class="c1"># add nodes and edges</span>
<span class="c1"># if type is 'i' just add nodea</span>
<span class="c1"># if type is a d connect to everything previous</span>
<span class="k">while</span> <span class="n">ci</span><span class="p">:</span>
<span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">node_type</span><span class="p">)</span> <span class="o">=</span> <span class="n">ci</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">if</span> <span class="n">node_type</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span> <span class="c1"># dominating type, connect to all existing nodes</span>
<span class="c1"># We use `for u in list(G):` instead of</span>
<span class="c1"># `for u in G:` because we edit the graph `G` in</span>
<span class="c1"># the loop. Hence using an iterator will result in</span>
<span class="c1"># `RuntimeError: dictionary changed size during iteration`</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</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="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">return</span> <span class="n">G</span>
<span class="k">def</span> <span class="nf">find_alternating_4_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns False if there aren't any alternating 4 cycles.</span>
<span class="sd"> Otherwise returns the cycle as [a,b,c,d] where (a,b)</span>
<span class="sd"> and (c,d) are edges and (a,c) and (b,d) are not.</span>
<span class="sd"> """</span>
<span class="k">for</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="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">():</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">():</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span> <span class="ow">and</span> <span class="n">u</span> <span class="o">!=</span> <span class="n">w</span><span class="p">:</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="ow">and</span> <span class="n">v</span> <span class="o">!=</span> <span class="n">x</span><span class="p">:</span>
<span class="k">return</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="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">]</span>
<span class="k">return</span> <span class="kc">False</span>
<div class="viewcode-block" id="find_threshold_graph"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.threshold.find_threshold_graph.html#networkx.algorithms.threshold.find_threshold_graph">[docs]</a><span class="k">def</span> <span class="nf">find_threshold_graph</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">create_using</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns a threshold subgraph that is close to largest in `G`.</span>
<span class="sd"> The threshold graph will contain the largest degree node in G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph instance</span>
<span class="sd"> An instance of `Graph`, or `MultiDiGraph`</span>
<span class="sd"> create_using : NetworkX graph class or `None` (default), optional</span>
<span class="sd"> Type of graph to use when constructing the threshold graph.</span>
<span class="sd"> If `None`, infer the appropriate graph type from the input.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> graph :</span>
<span class="sd"> A graph instance representing the threshold graph</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms.threshold import find_threshold_graph</span>
<span class="sd"> >>> G = nx.barbell_graph(3, 3)</span>
<span class="sd"> >>> T = find_threshold_graph(G)</span>
<span class="sd"> >>> T.nodes # may vary</span>
<span class="sd"> NodeView((7, 8, 5, 6))</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">threshold_graph</span><span class="p">(</span><span class="n">find_creation_sequence</span><span class="p">(</span><span class="n">G</span><span class="p">),</span> <span class="n">create_using</span><span class="p">)</span></div>
<span class="k">def</span> <span class="nf">find_creation_sequence</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Find a threshold subgraph that is close to largest in G.</span>
<span class="sd"> Returns the labeled creation sequence of that threshold graph.</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># get a local pointer to the working part of the graph</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">G</span>
<span class="k">while</span> <span class="n">H</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># get new degree sequence on subgraph</span>
<span class="n">dsdict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">H</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
<span class="n">ds</span> <span class="o">=</span> <span class="p">[(</span><span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dsdict</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
<span class="n">ds</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
<span class="c1"># Update threshold graph nodes</span>
<span class="k">if</span> <span class="n">ds</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># all are isolated</span>
<span class="n">cs</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">dsdict</span><span class="p">,</span> <span class="p">[</span><span class="s2">"i"</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="s2">"d"</span><span class="p">]))</span>
<span class="k">break</span> <span class="c1"># Done!</span>
<span class="c1"># pull off isolated nodes</span>
<span class="k">while</span> <span class="n">ds</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">iso</span><span class="p">)</span> <span class="o">=</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">iso</span><span class="p">,</span> <span class="s2">"i"</span><span class="p">))</span>
<span class="c1"># find new biggest node</span>
<span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">bigv</span><span class="p">)</span> <span class="o">=</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># add edges of star to t_g</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">bigv</span><span class="p">,</span> <span class="s2">"d"</span><span class="p">))</span>
<span class="c1"># form subgraph of neighbors of big node</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">H</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">bigv</span><span class="p">))</span>
<span class="n">cs</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="c1"># Properties of Threshold Graphs</span>
<span class="k">def</span> <span class="nf">triangles</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Compute number of triangles in the threshold graph with the</span>
<span class="sd"> given creation sequence.</span>
<span class="sd"> """</span>
<span class="c1"># shortcut algorithm that doesn't require computing number</span>
<span class="c1"># of triangles at each node.</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span> <span class="c1"># alias</span>
<span class="n">dr</span> <span class="o">=</span> <span class="n">cs</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s2">"d"</span><span class="p">)</span> <span class="c1"># number of d's in sequence</span>
<span class="n">ntri</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">*</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span> <span class="c1"># number of triangles in clique of nd d's</span>
<span class="c1"># now add dr choose 2 triangles for every 'i' in sequence where</span>
<span class="c1"># dr is the number of d's to the right of the current i</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">typ</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
<span class="k">if</span> <span class="n">typ</span> <span class="o">==</span> <span class="s2">"i"</span><span class="p">:</span>
<span class="n">ntri</span> <span class="o">+=</span> <span class="n">dr</span> <span class="o">*</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">dr</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">ntri</span>
<span class="k">def</span> <span class="nf">triangle_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return triangle sequence for the given threshold graph creation sequence.</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span>
<span class="n">seq</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">dr</span> <span class="o">=</span> <span class="n">cs</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s2">"d"</span><span class="p">)</span> <span class="c1"># number of d's to the right of the current pos</span>
<span class="n">dcur</span> <span class="o">=</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># number of triangles through a node of clique dr</span>
<span class="n">irun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># number of i's in the last run</span>
<span class="n">drun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># number of d's in the last run</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">sym</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
<span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">drun</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">tri</span> <span class="o">=</span> <span class="n">dcur</span> <span class="o">+</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">irun</span> <span class="c1"># new triangles at this d</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># cs[i]="i":</span>
<span class="k">if</span> <span class="n">prevsym</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span> <span class="c1"># new string of i's</span>
<span class="n">dcur</span> <span class="o">+=</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">irun</span> <span class="c1"># accumulate shared shortest paths</span>
<span class="n">irun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># reset i run counter</span>
<span class="n">dr</span> <span class="o">-=</span> <span class="n">drun</span> <span class="c1"># reduce number of d's to right</span>
<span class="n">drun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># reset d run counter</span>
<span class="n">irun</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">tri</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">*</span> <span class="p">(</span><span class="n">dr</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># new triangles at this i</span>
<span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tri</span><span class="p">)</span>
<span class="n">prevsym</span> <span class="o">=</span> <span class="n">sym</span>
<span class="k">return</span> <span class="n">seq</span>
<span class="k">def</span> <span class="nf">cluster_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return cluster sequence for the given threshold graph creation sequence.</span>
<span class="sd"> """</span>
<span class="n">triseq</span> <span class="o">=</span> <span class="n">triangle_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">degseq</span> <span class="o">=</span> <span class="n">degree_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">cseq</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">degseq</span><span class="p">):</span>
<span class="n">tri</span> <span class="o">=</span> <span class="n">triseq</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">if</span> <span class="n">deg</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># isolated vertex or single pair gets cc 0</span>
<span class="n">cseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">continue</span>
<span class="n">max_size</span> <span class="o">=</span> <span class="p">(</span><span class="n">deg</span> <span class="o">*</span> <span class="p">(</span><span class="n">deg</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">//</span> <span class="mi">2</span>
<span class="n">cseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tri</span> <span class="o">/</span> <span class="n">max_size</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cseq</span>
<span class="k">def</span> <span class="nf">degree_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return degree sequence for the threshold graph with the given</span>
<span class="sd"> creation sequence</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span> <span class="c1"># alias</span>
<span class="n">seq</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">rd</span> <span class="o">=</span> <span class="n">cs</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s2">"d"</span><span class="p">)</span> <span class="c1"># number of d to the right</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">sym</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
<span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">rd</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rd</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rd</span><span class="p">)</span>
<span class="k">return</span> <span class="n">seq</span>
<span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return the density of the graph with this creation_sequence.</span>
<span class="sd"> The density is the fraction of possible edges present.</span>
<span class="sd"> """</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">two_size</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">degree_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">))</span>
<span class="n">two_possible</span> <span class="o">=</span> <span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">den</span> <span class="o">=</span> <span class="n">two_size</span> <span class="o">/</span> <span class="n">two_possible</span>
<span class="k">return</span> <span class="n">den</span>
<span class="k">def</span> <span class="nf">degree_correlation</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return the degree-degree correlation over all edges.</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span>
<span class="n">s1</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># deg_i*deg_j</span>
<span class="n">s2</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># deg_i^2+deg_j^2</span>
<span class="n">s3</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># deg_i+deg_j</span>
<span class="n">m</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># number of edges</span>
<span class="n">rd</span> <span class="o">=</span> <span class="n">cs</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s2">"d"</span><span class="p">)</span> <span class="c1"># number of d nodes to the right</span>
<span class="n">rdi</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">sym</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span> <span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">]</span> <span class="c1"># index of "d"s</span>
<span class="n">ds</span> <span class="o">=</span> <span class="n">degree_sequence</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">sym</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
<span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">rdi</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Logic error in degree_correlation"</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">rdi</span><span class="p">)</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
<span class="n">rdi</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">degi</span> <span class="o">=</span> <span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">for</span> <span class="n">dj</span> <span class="ow">in</span> <span class="n">rdi</span><span class="p">:</span>
<span class="n">degj</span> <span class="o">=</span> <span class="n">ds</span><span class="p">[</span><span class="n">dj</span><span class="p">]</span>
<span class="n">s1</span> <span class="o">+=</span> <span class="n">degj</span> <span class="o">*</span> <span class="n">degi</span>
<span class="n">s2</span> <span class="o">+=</span> <span class="n">degi</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">degj</span><span class="o">**</span><span class="mi">2</span>
<span class="n">s3</span> <span class="o">+=</span> <span class="n">degi</span> <span class="o">+</span> <span class="n">degj</span>
<span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">denom</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">m</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">s3</span> <span class="o">*</span> <span class="n">s3</span>
<span class="n">numer</span> <span class="o">=</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">m</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">-</span> <span class="n">s3</span> <span class="o">*</span> <span class="n">s3</span>
<span class="k">if</span> <span class="n">denom</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">if</span> <span class="n">numer</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Zero Denominator but Numerator is </span><span class="si">{</span><span class="n">numer</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">numer</span> <span class="o">/</span> <span class="n">denom</span>
<span class="k">def</span> <span class="nf">shortest_path</span><span class="p">(</span><span class="n">creation_sequence</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="w"> </span><span class="sd">"""</span>
<span class="sd"> Find the shortest path between u and v in a</span>
<span class="sd"> threshold graph G with the given creation_sequence.</span>
<span class="sd"> For an unlabeled creation_sequence, the vertices</span>
<span class="sd"> u and v must be integers in (0,len(sequence)) referring</span>
<span class="sd"> to the position of the desired vertices in the sequence.</span>
<span class="sd"> For a labeled creation_sequence, u and v are labels of veritices.</span>
<span class="sd"> Use cs=creation_sequence(degree_sequence,with_labels=True)</span>
<span class="sd"> to convert a degree sequence to a creation sequence.</span>
<span class="sd"> Returns a list of vertices from u to v.</span>
<span class="sd"> Example: if they are neighbors, it returns [u,v]</span>
<span class="sd"> """</span>
<span class="c1"># Turn input sequence into a labeled creation sequence</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="n">i</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="nb">len</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">))]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="n">ci</span> <span class="o">=</span> <span class="n">uncompact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">ci</span><span class="p">[</span><span class="n">i</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="nb">len</span><span class="p">(</span><span class="n">ci</span><span class="p">))]</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Not a valid creation sequence type"</span><span class="p">)</span>
<span class="n">verts</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">cs</span><span class="p">]</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">verts</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Vertex </span><span class="si">{</span><span class="n">v</span><span class="si">}</span><span class="s2"> not in graph from creation_sequence"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">u</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">verts</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Vertex </span><span class="si">{</span><span class="n">u</span><span class="si">}</span><span class="s2"> not in graph from creation_sequence"</span><span class="p">)</span>
<span class="c1"># Done checking</span>
<span class="k">if</span> <span class="n">u</span> <span class="o">==</span> <span class="n">v</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">uindex</span> <span class="o">=</span> <span class="n">verts</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">vindex</span> <span class="o">=</span> <span class="n">verts</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">bigind</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">uindex</span><span class="p">,</span> <span class="n">vindex</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cs</span><span class="p">[</span><span class="n">bigind</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="k">return</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="c1"># must be that cs[bigind][1]=='i'</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">cs</span><span class="p">[</span><span class="n">bigind</span><span class="p">:]</span>
<span class="k">while</span> <span class="n">cs</span><span class="p">:</span>
<span class="n">vert</span> <span class="o">=</span> <span class="n">cs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">vert</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[</span><span class="n">u</span><span class="p">,</span> <span class="n">vert</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">]</span>
<span class="c1"># All after u are type 'i' so no connection</span>
<span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
<span class="k">def</span> <span class="nf">shortest_path_length</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return the shortest path length from indicated node to</span>
<span class="sd"> every other node for the threshold graph with the given</span>
<span class="sd"> creation sequence.</span>
<span class="sd"> Node is indicated by index i in creation_sequence unless</span>
<span class="sd"> creation_sequence is labeled in which case, i is taken to</span>
<span class="sd"> be the label of the node.</span>
<span class="sd"> Paths lengths in threshold graphs are at most 2.</span>
<span class="sd"> Length to unreachable nodes is set to -1.</span>
<span class="sd"> """</span>
<span class="c1"># Turn input sequence into a labeled creation sequence</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span> <span class="c1"># creation sequence</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span><span class="p">[:]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span> <span class="c1"># labeled creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">creation_sequence</span><span class="p">]</span>
<span class="n">i</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">creation_sequence</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span> <span class="c1"># compact creation sequence</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">uncompact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Not a valid creation sequence type"</span><span class="p">)</span>
<span class="c1"># Compute</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
<span class="n">spl</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">N</span> <span class="c1"># length 2 to every node</span>
<span class="n">spl</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># except self which is 0</span>
<span class="c1"># 1 for all d's to the right</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</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="n">N</span><span class="p">):</span>
<span class="k">if</span> <span class="n">cs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="n">spl</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">cs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span> <span class="c1"># 1 for all nodes to the left</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
<span class="n">spl</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1"># and -1 for any trailing i to indicate unreachable</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="k">if</span> <span class="n">cs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">spl</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
<span class="k">return</span> <span class="n">spl</span>
<span class="k">def</span> <span class="nf">betweenness_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">,</span> <span class="n">normalized</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return betweenness for the threshold graph with the given creation</span>
<span class="sd"> sequence. The result is unscaled. To scale the values</span>
<span class="sd"> to the iterval [0,1] divide by (n-1)*(n-2).</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="n">creation_sequence</span>
<span class="n">seq</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># betweenness</span>
<span class="n">lastchar</span> <span class="o">=</span> <span class="s2">"d"</span> <span class="c1"># first node is always a 'd'</span>
<span class="n">dr</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">cs</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s2">"d"</span><span class="p">))</span> <span class="c1"># number of d's to the right of curren pos</span>
<span class="n">irun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># number of i's in the last run</span>
<span class="n">drun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># number of d's in the last run</span>
<span class="n">dlast</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># betweenness of last d</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
<span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span> <span class="c1"># cs[i]=="d":</span>
<span class="c1"># betweennees = amt shared with eariler d's and i's</span>
<span class="c1"># + new isolated nodes covered</span>
<span class="c1"># + new paths to all previous nodes</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">dlast</span> <span class="o">+</span> <span class="p">(</span><span class="n">irun</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">irun</span> <span class="o">/</span> <span class="n">dr</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">irun</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">drun</span> <span class="o">-</span> <span class="n">irun</span><span class="p">)</span> <span class="o">/</span> <span class="n">dr</span>
<span class="n">drun</span> <span class="o">+=</span> <span class="mi">1</span> <span class="c1"># update counter</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># cs[i]="i":</span>
<span class="k">if</span> <span class="n">lastchar</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">:</span> <span class="c1"># if this is a new run of i's</span>
<span class="n">dlast</span> <span class="o">=</span> <span class="n">b</span> <span class="c1"># accumulate betweenness</span>
<span class="n">dr</span> <span class="o">-=</span> <span class="n">drun</span> <span class="c1"># update number of d's to the right</span>
<span class="n">drun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># reset d counter</span>
<span class="n">irun</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># reset i counter</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># isolated nodes have zero betweenness</span>
<span class="n">irun</span> <span class="o">+=</span> <span class="mi">1</span> <span class="c1"># add another i to the run</span>
<span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
<span class="n">lastchar</span> <span class="o">=</span> <span class="n">c</span>
<span class="c1"># normalize by the number of possible shortest paths</span>
<span class="k">if</span> <span class="n">normalized</span><span class="p">:</span>
<span class="n">order</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="p">((</span><span class="n">order</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">order</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">seq</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="o">*</span> <span class="n">scale</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">]</span>
<span class="k">return</span> <span class="n">seq</span>
<span class="k">def</span> <span class="nf">eigenvectors</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return a 2-tuple of Laplacian eigenvalues and eigenvectors</span>
<span class="sd"> for the threshold network with creation_sequence.</span>
<span class="sd"> The first value is a list of eigenvalues.</span>
<span class="sd"> The second value is a list of eigenvectors.</span>
<span class="sd"> The lists are in the same order so corresponding eigenvectors</span>
<span class="sd"> and eigenvalues are in the same position in the two lists.</span>
<span class="sd"> Notice that the order of the eigenvalues returned by eigenvalues(cs)</span>
<span class="sd"> may not correspond to the order of these eigenvectors.</span>
<span class="sd"> """</span>
<span class="n">ccs</span> <span class="o">=</span> <span class="n">make_compact</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ccs</span><span class="p">)</span>
<span class="n">vec</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">N</span>
<span class="n">val</span> <span class="o">=</span> <span class="n">vec</span><span class="p">[:]</span>
<span class="c1"># get number of type d nodes to the right (all for first node)</span>
<span class="n">dr</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ccs</span><span class="p">[::</span><span class="mi">2</span><span class="p">])</span>
<span class="n">nn</span> <span class="o">=</span> <span class="n">ccs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">vec</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span> <span class="o">*</span> <span class="n">N</span>
<span class="n">val</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">dr</span>
<span class="n">dr</span> <span class="o">-=</span> <span class="n">nn</span>
<span class="n">type_d</span> <span class="o">=</span> <span class="kc">True</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">dd</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">dd</span> <span class="o"><</span> <span class="n">nn</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">dd</span> <span class="o">*</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span>
<span class="n">vec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span> <span class="o">*</span> <span class="p">[</span><span class="o">-</span><span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">dd</span> <span class="o">*</span> <span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">dd</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ccs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">vec</span><span class="p">)</span>
<span class="k">for</span> <span class="n">nn</span> <span class="ow">in</span> <span class="n">ccs</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nn</span> <span class="o">*</span> <span class="n">i</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">nn</span><span class="p">))</span>
<span class="n">vec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span> <span class="o">*</span> <span class="p">[</span><span class="o">-</span><span class="n">nn</span> <span class="o">*</span> <span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="n">nn</span> <span class="o">*</span> <span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="n">nn</span><span class="p">)</span>
<span class="c1"># find eigenvalue</span>
<span class="n">type_d</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">type_d</span>
<span class="k">if</span> <span class="n">type_d</span><span class="p">:</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">dr</span>
<span class="n">dr</span> <span class="o">-=</span> <span class="n">nn</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">dr</span>
<span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
<span class="n">st</span> <span class="o">=</span> <span class="n">i</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">dd</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">dd</span> <span class="o"><</span> <span class="n">nn</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">st</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">*</span> <span class="n">dd</span><span class="p">)</span>
<span class="n">vec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">st</span> <span class="o">+</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">st</span><span class="p">)</span> <span class="o">*</span> <span class="p">[</span><span class="o">-</span><span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">dd</span> <span class="o">*</span> <span class="n">scale</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">dd</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">vec</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">spectral_projection</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">eigenpairs</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns the coefficients of each eigenvector</span>
<span class="sd"> in a projection of the vector u onto the normalized</span>
<span class="sd"> eigenvectors which are contained in eigenpairs.</span>
<span class="sd"> eigenpairs should be a list of two objects. The</span>
<span class="sd"> first is a list of eigenvalues and the second a list</span>
<span class="sd"> of eigenvectors. The eigenvectors should be lists.</span>
<span class="sd"> There's not a lot of error checking on lengths of</span>
<span class="sd"> arrays, etc. so be careful.</span>
<span class="sd"> """</span>
<span class="n">coeff</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">evect</span> <span class="o">=</span> <span class="n">eigenpairs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="n">ev</span> <span class="ow">in</span> <span class="n">evect</span><span class="p">:</span>
<span class="n">c</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">evv</span> <span class="o">*</span> <span class="n">uv</span> <span class="k">for</span> <span class="p">(</span><span class="n">evv</span><span class="p">,</span> <span class="n">uv</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">ev</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span>
<span class="n">coeff</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="k">return</span> <span class="n">coeff</span>
<span class="k">def</span> <span class="nf">eigenvalues</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Return sequence of eigenvalues of the Laplacian of the threshold</span>
<span class="sd"> graph for the given creation_sequence.</span>
<span class="sd"> Based on the Ferrer's diagram method. The spectrum is integral</span>
<span class="sd"> and is the conjugate of the degree sequence.</span>
<span class="sd"> See::</span>
<span class="sd"> @Article{degree-merris-1994,</span>
<span class="sd"> author = {Russel Merris},</span>
<span class="sd"> title = {Degree maximal graphs are Laplacian integral},</span>
<span class="sd"> journal = {Linear Algebra Appl.},</span>
<span class="sd"> year = {1994},</span>
<span class="sd"> volume = {199},</span>
<span class="sd"> pages = {381--389},</span>
<span class="sd"> }</span>
<span class="sd"> """</span>
<span class="n">degseq</span> <span class="o">=</span> <span class="n">degree_sequence</span><span class="p">(</span><span class="n">creation_sequence</span><span class="p">)</span>
<span class="n">degseq</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
<span class="n">eiglist</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># zero is always one eigenvalue</span>
<span class="n">eig</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">row</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">degseq</span><span class="p">)</span>
<span class="n">bigdeg</span> <span class="o">=</span> <span class="n">degseq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">while</span> <span class="n">row</span><span class="p">:</span>
<span class="k">if</span> <span class="n">bigdeg</span> <span class="o"><</span> <span class="n">row</span><span class="p">:</span>
<span class="n">eiglist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">eig</span><span class="p">)</span>
<span class="n">row</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">eig</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">degseq</span><span class="p">:</span>
<span class="n">bigdeg</span> <span class="o">=</span> <span class="n">degseq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">bigdeg</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">eiglist</span>
<span class="c1"># Threshold graph creation routines</span>
<span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">random_threshold_sequence</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Create a random threshold sequence of size n.</span>
<span class="sd"> A creation sequence is built by randomly choosing d's with</span>
<span class="sd"> probabiliy p and i's with probability 1-p.</span>
<span class="sd"> s=nx.random_threshold_sequence(10,0.5)</span>
<span class="sd"> returns a threshold sequence of length 10 with equal</span>
<span class="sd"> probably of an i or a d at each position.</span>
<span class="sd"> A "random" threshold graph can be built with</span>
<span class="sd"> G=nx.threshold_graph(s)</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="mi">0</span> <span class="o"><=</span> <span class="n">p</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"p must be in [0,1]"</span><span class="p">)</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"d"</span><span class="p">]</span> <span class="c1"># threshold sequences always start with a d</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="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">if</span> <span class="n">seed</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o"><</span> <span class="n">p</span><span class="p">:</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s2">"d"</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s2">"i"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="c1"># maybe *_d_threshold_sequence routines should</span>
<span class="c1"># be (or be called from) a single routine with a more descriptive name</span>
<span class="c1"># and a keyword parameter?</span>
<span class="k">def</span> <span class="nf">right_d_threshold_sequence</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Create a skewed threshold graph with a given number</span>
<span class="sd"> of vertices (n) and a given number of edges (m).</span>
<span class="sd"> The routine returns an unlabeled creation sequence</span>
<span class="sd"> for the threshold graph.</span>
<span class="sd"> FIXME: describe algorithm</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"d"</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s2">"i"</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># create sequence with n insolated nodes</span>
<span class="c1"># m <n : not enough edges, make disconnected</span>
<span class="k">if</span> <span class="n">m</span> <span class="o"><</span> <span class="n">n</span><span class="p">:</span>
<span class="n">cs</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="c1"># too many edges</span>
<span class="k">if</span> <span class="n">m</span> <span class="o">></span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"Too many edges for this many nodes."</span><span class="p">)</span>
<span class="c1"># connected case m >n-1</span>
<span class="n">ind</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<span class="nb">sum</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">while</span> <span class="nb">sum</span> <span class="o"><</span> <span class="n">m</span><span class="p">:</span>
<span class="n">cs</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="n">ind</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="nb">sum</span> <span class="o">+=</span> <span class="n">ind</span>
<span class="n">ind</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="p">(</span><span class="nb">sum</span> <span class="o">-</span> <span class="n">ind</span><span class="p">)</span>
<span class="n">cs</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="k">def</span> <span class="nf">left_d_threshold_sequence</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Create a skewed threshold graph with a given number</span>
<span class="sd"> of vertices (n) and a given number of edges (m).</span>
<span class="sd"> The routine returns an unlabeled creation sequence</span>
<span class="sd"> for the threshold graph.</span>
<span class="sd"> FIXME: describe algorithm</span>
<span class="sd"> """</span>
<span class="n">cs</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"d"</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s2">"i"</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># create sequence with n insolated nodes</span>
<span class="c1"># m <n : not enough edges, make disconnected</span>
<span class="k">if</span> <span class="n">m</span> <span class="o"><</span> <span class="n">n</span><span class="p">:</span>
<span class="n">cs</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="c1"># too many edges</span>
<span class="k">if</span> <span class="n">m</span> <span class="o">></span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"Too many edges for this many nodes."</span><span class="p">)</span>
<span class="c1"># Connected case when M>N-1</span>
<span class="n">cs</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="nb">sum</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<span class="n">ind</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="nb">sum</span> <span class="o"><</span> <span class="n">m</span><span class="p">:</span>
<span class="n">cs</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="nb">sum</span> <span class="o">+=</span> <span class="n">ind</span>
<span class="n">ind</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="nb">sum</span> <span class="o">></span> <span class="n">m</span><span class="p">:</span> <span class="c1"># be sure not to change the first vertex</span>
<span class="n">cs</span><span class="p">[</span><span class="nb">sum</span> <span class="o">-</span> <span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"i"</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">swap_d</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">p_split</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">p_combine</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Perform a "swap" operation on a threshold sequence.</span>
<span class="sd"> The swap preserves the number of nodes and edges</span>
<span class="sd"> in the graph for the given sequence.</span>
<span class="sd"> The resulting sequence is still a threshold sequence.</span>
<span class="sd"> Perform one split and one combine operation on the</span>
<span class="sd"> 'd's of a creation sequence for a threshold graph.</span>
<span class="sd"> This operation maintains the number of nodes and edges</span>
<span class="sd"> in the graph, but shifts the edges from node to node</span>
<span class="sd"> maintaining the threshold quality of the graph.</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> """</span>
<span class="c1"># preprocess the creation sequence</span>
<span class="n">dlist</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">node_type</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cs</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="k">if</span> <span class="n">node_type</span> <span class="o">==</span> <span class="s2">"d"</span><span class="p">]</span>
<span class="c1"># split</span>
<span class="k">if</span> <span class="n">seed</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o"><</span> <span class="n">p_split</span><span class="p">:</span>
<span class="n">choice</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">dlist</span><span class="p">)</span>
<span class="n">split_to</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span>
<span class="n">flip_side</span> <span class="o">=</span> <span class="n">choice</span> <span class="o">-</span> <span class="n">split_to</span>
<span class="k">if</span> <span class="n">split_to</span> <span class="o">!=</span> <span class="n">flip_side</span> <span class="ow">and</span> <span class="n">cs</span><span class="p">[</span><span class="n">split_to</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"i"</span> <span class="ow">and</span> <span class="n">cs</span><span class="p">[</span><span class="n">flip_side</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"i"</span><span class="p">:</span>
<span class="n">cs</span><span class="p">[</span><span class="n">choice</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"i"</span>
<span class="n">cs</span><span class="p">[</span><span class="n">split_to</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="n">cs</span><span class="p">[</span><span class="n">flip_side</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="n">dlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
<span class="c1"># don't add or combine may reverse this action</span>
<span class="c1"># dlist.extend([split_to,flip_side])</span>
<span class="c1"># print >>sys.stderr,"split at %s to %s and %s"%(choice,split_to,flip_side)</span>
<span class="c1"># combine</span>
<span class="k">if</span> <span class="n">seed</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o"><</span> <span class="n">p_combine</span> <span class="ow">and</span> <span class="n">dlist</span><span class="p">:</span>
<span class="n">first_choice</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">dlist</span><span class="p">)</span>
<span class="n">second_choice</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">dlist</span><span class="p">)</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">first_choice</span> <span class="o">+</span> <span class="n">second_choice</span>
<span class="k">if</span> <span class="n">target</span> <span class="o">>=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span> <span class="ow">or</span> <span class="n">cs</span><span class="p">[</span><span class="n">target</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"d"</span> <span class="ow">or</span> <span class="n">first_choice</span> <span class="o">==</span> <span class="n">second_choice</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cs</span>
<span class="c1"># OK to combine</span>
<span class="n">cs</span><span class="p">[</span><span class="n">first_choice</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"i"</span>
<span class="n">cs</span><span class="p">[</span><span class="n">second_choice</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"i"</span>
<span class="n">cs</span><span class="p">[</span><span class="n">target</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"d"</span>
<span class="c1"># print >>sys.stderr,"combine %s and %s to make %s."%(first_choice,second_choice,target)</span>
<span class="k">return</span> <span class="n">cs</span>
</pre></div>
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