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<h1>Source code for networkx.algorithms.smallworld</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Functions for estimating the small-world-ness of graphs.</span>
<span class="sd">A small world network is characterized by a small average shortest path length,</span>
<span class="sd">and a large clustering coefficient.</span>
<span class="sd">Small-worldness is commonly measured with the coefficient sigma or omega.</span>
<span class="sd">Both coefficients compare the average clustering coefficient and shortest path</span>
<span class="sd">length of a given graph against the same quantities for an equivalent random</span>
<span class="sd">or lattice graph.</span>
<span class="sd">For more information, see the Wikipedia article on small-world network [1]_.</span>
<span class="sd">.. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network</span>
<span class="sd">"""</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">not_implemented_for</span><span class="p">,</span> <span class="n">py_random_state</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"random_reference"</span><span class="p">,</span> <span class="s2">"lattice_reference"</span><span class="p">,</span> <span class="s2">"sigma"</span><span class="p">,</span> <span class="s2">"omega"</span><span class="p">]</span>
<div class="viewcode-block" id="random_reference"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.smallworld.random_reference.html#networkx.algorithms.smallworld.random_reference">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">random_reference</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">connectivity</span><span class="o">=</span><span class="kc">True</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="sd">"""Compute a random graph by swapping edges of a given graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> An undirected graph with 4 or more nodes.</span>
<span class="sd"> niter : integer (optional, default=1)</span>
<span class="sd"> An edge is rewired approximately `niter` times.</span>
<span class="sd"> connectivity : boolean (optional, default=True)</span>
<span class="sd"> When True, ensure connectivity for the randomized 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"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> G : graph</span>
<span class="sd"> The randomized graph.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If there are fewer than 4 nodes or 2 edges in `G`</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation is adapted from the algorithm by Maslov and Sneppen</span>
<span class="sd"> (2002) [1]_.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Maslov, Sergei, and Kim Sneppen.</span>
<span class="sd"> "Specificity and stability in topology of protein networks."</span>
<span class="sd"> Science 296.5569 (2002): 910-913.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o"><</span> <span class="mi">4</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">"Graph has fewer than four nodes."</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">edges</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="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Graph has fewer that 2 edges"</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">cumulative_distribution</span><span class="p">,</span> <span class="n">discrete_sequence</span>
<span class="n">local_conn</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">connectivity</span><span class="o">.</span><span class="n">local_edge_connectivity</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">keys</span><span class="p">,</span> <span class="n">degrees</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span> <span class="c1"># keys, degree</span>
<span class="n">cdf</span> <span class="o">=</span> <span class="n">cumulative_distribution</span><span class="p">(</span><span class="n">degrees</span><span class="p">)</span> <span class="c1"># cdf of degree</span>
<span class="n">nnodes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">nedges</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">number_of_edges</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">niter</span> <span class="o">=</span> <span class="n">niter</span> <span class="o">*</span> <span class="n">nedges</span>
<span class="n">ntries</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nnodes</span> <span class="o">*</span> <span class="n">nedges</span> <span class="o">/</span> <span class="p">(</span><span class="n">nnodes</span> <span class="o">*</span> <span class="p">(</span><span class="n">nnodes</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="n">swapcount</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">n</span> <span class="o"><</span> <span class="n">ntries</span><span class="p">:</span>
<span class="c1"># pick two random edges without creating edge list</span>
<span class="c1"># choose source node indices from discrete distribution</span>
<span class="p">(</span><span class="n">ai</span><span class="p">,</span> <span class="n">ci</span><span class="p">)</span> <span class="o">=</span> <span class="n">discrete_sequence</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">cdistribution</span><span class="o">=</span><span class="n">cdf</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ai</span> <span class="o">==</span> <span class="n">ci</span><span class="p">:</span>
<span class="k">continue</span> <span class="c1"># same source, skip</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">ai</span><span class="p">]</span> <span class="c1"># convert index to label</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">ci</span><span class="p">]</span>
<span class="c1"># choose target uniformly from neighbors</span>
<span class="n">b</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">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
<span class="n">d</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">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">c</span><span class="p">)))</span>
<span class="k">if</span> <span class="n">b</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="ow">or</span> <span class="n">d</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]:</span>
<span class="k">continue</span> <span class="c1"># all vertices should be different</span>
<span class="c1"># don't create parallel edges</span>
<span class="k">if</span> <span class="p">(</span><span class="n">d</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">a</span><span class="p">])</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">c</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">a</span><span class="p">,</span> <span class="n">d</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">c</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="c1"># Check if the graph is still connected</span>
<span class="k">if</span> <span class="n">connectivity</span> <span class="ow">and</span> <span class="n">local_conn</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># Not connected, revert the swap</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">b</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">a</span><span class="p">,</span> <span class="n">b</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">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">swapcount</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">break</span>
<span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">G</span></div>
<div class="viewcode-block" id="lattice_reference"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.smallworld.lattice_reference.html#networkx.algorithms.smallworld.lattice_reference">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">lattice_reference</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">connectivity</span><span class="o">=</span><span class="kc">True</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="sd">"""Latticize the given graph by swapping edges.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> An undirected graph.</span>
<span class="sd"> niter : integer (optional, default=1)</span>
<span class="sd"> An edge is rewired approximatively niter times.</span>
<span class="sd"> D : numpy.array (optional, default=None)</span>
<span class="sd"> Distance to the diagonal matrix.</span>
<span class="sd"> connectivity : boolean (optional, default=True)</span>
<span class="sd"> Ensure connectivity for the latticized graph when set to True.</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"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> G : graph</span>
<span class="sd"> The latticized graph.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If there are fewer than 4 nodes or 2 edges in `G`</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation is adapted from the algorithm by Sporns et al. [1]_.</span>
<span class="sd"> which is inspired from the original work by Maslov and Sneppen(2002) [2]_.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Sporns, Olaf, and Jonathan D. Zwi.</span>
<span class="sd"> "The small world of the cerebral cortex."</span>
<span class="sd"> Neuroinformatics 2.2 (2004): 145-162.</span>
<span class="sd"> .. [2] Maslov, Sergei, and Kim Sneppen.</span>
<span class="sd"> "Specificity and stability in topology of protein networks."</span>
<span class="sd"> Science 296.5569 (2002): 910-913.</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">cumulative_distribution</span><span class="p">,</span> <span class="n">discrete_sequence</span>
<span class="n">local_conn</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">connectivity</span><span class="o">.</span><span class="n">local_edge_connectivity</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o"><</span> <span class="mi">4</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">"Graph has fewer than four nodes."</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">edges</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="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Graph has fewer that 2 edges"</span><span class="p">)</span>
<span class="c1"># Instead of choosing uniformly at random from a generated edge list,</span>
<span class="c1"># this algorithm chooses nonuniformly from the set of nodes with</span>
<span class="c1"># probability weighted by degree.</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">keys</span><span class="p">,</span> <span class="n">degrees</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span> <span class="c1"># keys, degree</span>
<span class="n">cdf</span> <span class="o">=</span> <span class="n">cumulative_distribution</span><span class="p">(</span><span class="n">degrees</span><span class="p">)</span> <span class="c1"># cdf of degree</span>
<span class="n">nnodes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">nedges</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">number_of_edges</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nnodes</span><span class="p">,</span> <span class="n">nnodes</span><span class="p">))</span>
<span class="n">un</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">nnodes</span><span class="p">)</span>
<span class="n">um</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nnodes</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="n">u</span> <span class="o">=</span> <span class="n">np</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="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">un</span> <span class="o"><</span> <span class="n">um</span><span class="p">,</span> <span class="n">un</span><span class="p">,</span> <span class="n">um</span><span class="p">))</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">nnodes</span> <span class="o">/</span> <span class="mi">2</span><span class="p">))):</span>
<span class="n">D</span><span class="p">[</span><span class="n">nnodes</span> <span class="o">-</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">v</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">:],</span> <span class="n">u</span><span class="p">[:</span> <span class="n">v</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="n">v</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">D</span><span class="p">[</span><span class="n">nnodes</span> <span class="o">-</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">niter</span> <span class="o">=</span> <span class="n">niter</span> <span class="o">*</span> <span class="n">nedges</span>
<span class="c1"># maximal number of rewiring attempts per 'niter'</span>
<span class="n">max_attempts</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nnodes</span> <span class="o">*</span> <span class="n">nedges</span> <span class="o">/</span> <span class="p">(</span><span class="n">nnodes</span> <span class="o">*</span> <span class="p">(</span><span class="n">nnodes</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">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">n</span> <span class="o"><</span> <span class="n">max_attempts</span><span class="p">:</span>
<span class="c1"># pick two random edges without creating edge list</span>
<span class="c1"># choose source node indices from discrete distribution</span>
<span class="p">(</span><span class="n">ai</span><span class="p">,</span> <span class="n">ci</span><span class="p">)</span> <span class="o">=</span> <span class="n">discrete_sequence</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">cdistribution</span><span class="o">=</span><span class="n">cdf</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ai</span> <span class="o">==</span> <span class="n">ci</span><span class="p">:</span>
<span class="k">continue</span> <span class="c1"># same source, skip</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">ai</span><span class="p">]</span> <span class="c1"># convert index to label</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">ci</span><span class="p">]</span>
<span class="c1"># choose target uniformly from neighbors</span>
<span class="n">b</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">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
<span class="n">d</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">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">c</span><span class="p">)))</span>
<span class="n">bi</span> <span class="o">=</span> <span class="n">keys</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">di</span> <span class="o">=</span> <span class="n">keys</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="k">if</span> <span class="n">b</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="ow">or</span> <span class="n">d</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]:</span>
<span class="k">continue</span> <span class="c1"># all vertices should be different</span>
<span class="c1"># don't create parallel edges</span>
<span class="k">if</span> <span class="p">(</span><span class="n">d</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">a</span><span class="p">])</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">c</span><span class="p">]):</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">ai</span><span class="p">,</span> <span class="n">bi</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">ci</span><span class="p">,</span> <span class="n">di</span><span class="p">]</span> <span class="o">>=</span> <span class="n">D</span><span class="p">[</span><span class="n">ai</span><span class="p">,</span> <span class="n">ci</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">bi</span><span class="p">,</span> <span class="n">di</span><span class="p">]:</span>
<span class="c1"># only swap if we get closer to the diagonal</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">d</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">c</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="c1"># Check if the graph is still connected</span>
<span class="k">if</span> <span class="n">connectivity</span> <span class="ow">and</span> <span class="n">local_conn</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># Not connected, revert the swap</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">b</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">a</span><span class="p">,</span> <span class="n">b</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">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">G</span></div>
<div class="viewcode-block" id="sigma"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.smallworld.sigma.html#networkx.algorithms.smallworld.sigma">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">sigma</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">nrand</span><span class="o">=</span><span class="mi">10</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="sd">"""Returns the small-world coefficient (sigma) of the given graph.</span>
<span class="sd"> The small-world coefficient is defined as:</span>
<span class="sd"> sigma = C/Cr / L/Lr</span>
<span class="sd"> where C and L are respectively the average clustering coefficient and</span>
<span class="sd"> average shortest path length of G. Cr and Lr are respectively the average</span>
<span class="sd"> clustering coefficient and average shortest path length of an equivalent</span>
<span class="sd"> random graph.</span>
<span class="sd"> A graph is commonly classified as small-world if sigma>1.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> An undirected graph.</span>
<span class="sd"> niter : integer (optional, default=100)</span>
<span class="sd"> Approximate number of rewiring per edge to compute the equivalent</span>
<span class="sd"> random graph.</span>
<span class="sd"> nrand : integer (optional, default=10)</span>
<span class="sd"> Number of random graphs generated to compute the average clustering</span>
<span class="sd"> coefficient (Cr) and average shortest path length (Lr).</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"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> sigma : float</span>
<span class="sd"> The small-world coefficient of G.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation is adapted from Humphries et al. [1]_ [2]_.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] The brainstem reticular formation is a small-world, not scale-free,</span>
<span class="sd"> network M. D. Humphries, K. Gurney and T. J. Prescott,</span>
<span class="sd"> Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354.</span>
<span class="sd"> .. [2] Humphries and Gurney (2008).</span>
<span class="sd"> "Network 'Small-World-Ness': A Quantitative Method for Determining</span>
<span class="sd"> Canonical Network Equivalence".</span>
<span class="sd"> PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051.</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># Compute the mean clustering coefficient and average shortest path length</span>
<span class="c1"># for an equivalent random graph</span>
<span class="n">randMetrics</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"C"</span><span class="p">:</span> <span class="p">[],</span> <span class="s2">"L"</span><span class="p">:</span> <span class="p">[]}</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nrand</span><span class="p">):</span>
<span class="n">Gr</span> <span class="o">=</span> <span class="n">random_reference</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="n">randMetrics</span><span class="p">[</span><span class="s2">"C"</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">transitivity</span><span class="p">(</span><span class="n">Gr</span><span class="p">))</span>
<span class="n">randMetrics</span><span class="p">[</span><span class="s2">"L"</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">average_shortest_path_length</span><span class="p">(</span><span class="n">Gr</span><span class="p">))</span>
<span class="n">C</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">transitivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">average_shortest_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">Cr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">randMetrics</span><span class="p">[</span><span class="s2">"C"</span><span class="p">])</span>
<span class="n">Lr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">randMetrics</span><span class="p">[</span><span class="s2">"L"</span><span class="p">])</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="p">(</span><span class="n">C</span> <span class="o">/</span> <span class="n">Cr</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">L</span> <span class="o">/</span> <span class="n">Lr</span><span class="p">)</span>
<span class="k">return</span> <span class="n">sigma</span></div>
<div class="viewcode-block" id="omega"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.smallworld.omega.html#networkx.algorithms.smallworld.omega">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">omega</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">nrand</span><span class="o">=</span><span class="mi">10</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="sd">"""Returns the small-world coefficient (omega) of a graph</span>
<span class="sd"> The small-world coefficient of a graph G is:</span>
<span class="sd"> omega = Lr/L - C/Cl</span>
<span class="sd"> where C and L are respectively the average clustering coefficient and</span>
<span class="sd"> average shortest path length of G. Lr is the average shortest path length</span>
<span class="sd"> of an equivalent random graph and Cl is the average clustering coefficient</span>
<span class="sd"> of an equivalent lattice graph.</span>
<span class="sd"> The small-world coefficient (omega) measures how much G is like a lattice</span>
<span class="sd"> or a random graph. Negative values mean G is similar to a lattice whereas</span>
<span class="sd"> positive values mean G is a random graph.</span>
<span class="sd"> Values close to 0 mean that G has small-world characteristics.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> An undirected graph.</span>
<span class="sd"> niter: integer (optional, default=5)</span>
<span class="sd"> Approximate number of rewiring per edge to compute the equivalent</span>
<span class="sd"> random graph.</span>
<span class="sd"> nrand: integer (optional, default=10)</span>
<span class="sd"> Number of random graphs generated to compute the maximal clustering</span>
<span class="sd"> coefficient (Cr) and average shortest path length (Lr).</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"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> omega : float</span>
<span class="sd"> The small-world coefficient (omega)</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation is adapted from the algorithm by Telesford et al. [1]_.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011).</span>
<span class="sd"> "The Ubiquity of Small-World Networks".</span>
<span class="sd"> Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451.</span>
<span class="sd"> doi:10.1089/brain.2011.0038.</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># Compute the mean clustering coefficient and average shortest path length</span>
<span class="c1"># for an equivalent random graph</span>
<span class="n">randMetrics</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"C"</span><span class="p">:</span> <span class="p">[],</span> <span class="s2">"L"</span><span class="p">:</span> <span class="p">[]}</span>
<span class="c1"># Calculate initial average clustering coefficient which potentially will</span>
<span class="c1"># get replaced by higher clustering coefficients from generated lattice</span>
<span class="c1"># reference graphs</span>
<span class="n">Cl</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">average_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">niter_lattice_reference</span> <span class="o">=</span> <span class="n">niter</span>
<span class="n">niter_random_reference</span> <span class="o">=</span> <span class="n">niter</span> <span class="o">*</span> <span class="mi">2</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nrand</span><span class="p">):</span>
<span class="c1"># Generate random graph</span>
<span class="n">Gr</span> <span class="o">=</span> <span class="n">random_reference</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter_random_reference</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="n">randMetrics</span><span class="p">[</span><span class="s2">"L"</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">average_shortest_path_length</span><span class="p">(</span><span class="n">Gr</span><span class="p">))</span>
<span class="c1"># Generate lattice graph</span>
<span class="n">Gl</span> <span class="o">=</span> <span class="n">lattice_reference</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter_lattice_reference</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="c1"># Replace old clustering coefficient, if clustering is higher in</span>
<span class="c1"># generated lattice reference</span>
<span class="n">Cl_temp</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">average_clustering</span><span class="p">(</span><span class="n">Gl</span><span class="p">)</span>
<span class="k">if</span> <span class="n">Cl_temp</span> <span class="o">></span> <span class="n">Cl</span><span class="p">:</span>
<span class="n">Cl</span> <span class="o">=</span> <span class="n">Cl_temp</span>
<span class="n">C</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">average_clustering</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">average_shortest_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">Lr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">randMetrics</span><span class="p">[</span><span class="s2">"L"</span><span class="p">])</span>
<span class="n">omega</span> <span class="o">=</span> <span class="p">(</span><span class="n">Lr</span> <span class="o">/</span> <span class="n">L</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">C</span> <span class="o">/</span> <span class="n">Cl</span><span class="p">)</span>
<span class="k">return</span> <span class="n">omega</span></div>
</pre></div>
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