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<h1>Source code for networkx.algorithms.community.modularity_max</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Functions for detecting communities based on modularity."""</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</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.algorithms.community.quality</span> <span class="kn">import</span> <span class="n">modularity</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="kn">from</span> <span class="nn">networkx.utils.mapped_queue</span> <span class="kn">import</span> <span class="n">MappedQueue</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"greedy_modularity_communities"</span><span class="p">,</span>
<span class="s2">"naive_greedy_modularity_communities"</span><span class="p">,</span>
<span class="p">]</span>
<span class="k">def</span> <span class="nf">_greedy_modularity_communities_generator</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">resolution</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Yield community partitions of G and the modularity change at each step.</span>
<span class="sd"> This function performs Clauset-Newman-Moore greedy modularity maximization [2]_</span>
<span class="sd"> At each step of the process it yields the change in modularity that will occur in</span>
<span class="sd"> the next step followed by yielding the new community partition after that step.</span>
<span class="sd"> Greedy modularity maximization begins with each node in its own community</span>
<span class="sd"> and repeatedly joins the pair of communities that lead to the largest</span>
<span class="sd"> modularity until one community contains all nodes (the partition has one set).</span>
<span class="sd"> This function maximizes the generalized modularity, where `resolution`</span>
<span class="sd"> is the resolution parameter, often expressed as $\gamma$.</span>
<span class="sd"> See :func:`~networkx.algorithms.community.quality.modularity`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or None, optional (default=None)</span>
<span class="sd"> The name of an edge attribute that holds the numerical value used</span>
<span class="sd"> as a weight. If None, then each edge has weight 1.</span>
<span class="sd"> The degree is the sum of the edge weights adjacent to the node.</span>
<span class="sd"> resolution : float (default=1)</span>
<span class="sd"> If resolution is less than 1, modularity favors larger communities.</span>
<span class="sd"> Greater than 1 favors smaller communities.</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> Alternating yield statements produce the following two objects:</span>
<span class="sd"> communities: dict_values</span>
<span class="sd"> A dict_values of frozensets of nodes, one for each community.</span>
<span class="sd"> This represents a partition of the nodes of the graph into communities.</span>
<span class="sd"> The first yield is the partition with each node in its own community.</span>
<span class="sd"> dq: float</span>
<span class="sd"> The change in modularity when merging the next two communities</span>
<span class="sd"> that leads to the largest modularity.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> modularity</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Newman, M. E. J. "Networks: An Introduction", page 224</span>
<span class="sd"> Oxford University Press 2011.</span>
<span class="sd"> .. [2] Clauset, A., Newman, M. E., & Moore, C.</span>
<span class="sd"> "Finding community structure in very large networks."</span>
<span class="sd"> Physical Review E 70(6), 2004.</span>
<span class="sd"> .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community</span>
<span class="sd"> Detection" Phys. Rev. E74, 2006.</span>
<span class="sd"> .. [4] Newman, M. E. J."Analysis of weighted networks"</span>
<span class="sd"> Physical Review E 70(5 Pt 2):056131, 2004.</span>
<span class="sd"> """</span>
<span class="n">directed</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">()</span>
<span class="n">N</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span>
<span class="c1"># Count edges (or the sum of edge-weights for weighted graphs)</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">weight</span><span class="p">)</span>
<span class="n">q0</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">m</span>
<span class="c1"># Calculate degrees (notation from the papers)</span>
<span class="c1"># a : the fraction of (weighted) out-degree for each node</span>
<span class="c1"># b : the fraction of (weighted) in-degree for each node</span>
<span class="k">if</span> <span class="n">directed</span><span class="p">:</span>
<span class="n">a</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span><span class="p">:</span> <span class="n">deg_out</span> <span class="o">*</span> <span class="n">q0</span> <span class="k">for</span> <span class="n">node</span><span class="p">,</span> <span class="n">deg_out</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">(</span><span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)}</span>
<span class="n">b</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span><span class="p">:</span> <span class="n">deg_in</span> <span class="o">*</span> <span class="n">q0</span> <span class="k">for</span> <span class="n">node</span><span class="p">,</span> <span class="n">deg_in</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">(</span><span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)}</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">b</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span><span class="p">:</span> <span class="n">deg</span> <span class="o">*</span> <span class="n">q0</span> <span class="o">*</span> <span class="mf">0.5</span> <span class="k">for</span> <span class="n">node</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)}</span>
<span class="c1"># this preliminary step collects the edge weights for each node pair</span>
<span class="c1"># It handles multigraph and digraph and works fine for graph.</span>
<span class="n">dq_dict</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">float</span><span class="p">))</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">wt</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="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="mi">1</span><span class="p">):</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">continue</span>
<span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">+=</span> <span class="n">wt</span>
<span class="n">dq_dict</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">u</span><span class="p">]</span> <span class="o">+=</span> <span class="n">wt</span>
<span class="c1"># now scale and subtract the expected edge-weights term</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">dq_dict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">wt</span> <span class="ow">in</span> <span class="n">nbrdict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">wt</span> <span class="o">-</span> <span class="n">resolution</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="n">b</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">*</span> <span class="n">a</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="c1"># Use -dq to get a max_heap instead of a min_heap</span>
<span class="c1"># dq_heap holds a heap for each node's neighbors</span>
<span class="n">dq_heap</span> <span class="o">=</span> <span class="p">{</span><span class="n">u</span><span class="p">:</span> <span class="n">MappedQueue</span><span class="p">({(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span> <span class="o">-</span><span class="n">dq</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">dq</span> <span class="ow">in</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()})</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span>
<span class="c1"># H -> all_dq_heap holds a heap with the best items for each node</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">MappedQueue</span><span class="p">([</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="o">></span> <span class="mi">0</span><span class="p">])</span>
<span class="c1"># Initialize single-node communities</span>
<span class="n">communities</span> <span class="o">=</span> <span class="p">{</span><span class="n">n</span><span class="p">:</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">n</span><span class="p">])</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span>
<span class="k">yield</span> <span class="n">communities</span><span class="o">.</span><span class="n">values</span><span class="p">()</span>
<span class="c1"># Merge the two communities that lead to the largest modularity</span>
<span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="c1"># Find best merge</span>
<span class="c1"># Remove from heap of row maxes</span>
<span class="c1"># Ties will be broken by choosing the pair with lowest min community id</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">negdq</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="n">H</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">dq</span> <span class="o">=</span> <span class="o">-</span><span class="n">negdq</span>
<span class="k">yield</span> <span class="n">dq</span>
<span class="c1"># Remove best merge from row u heap</span>
<span class="n">dq_heap</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># Push new row max onto H</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">u</span><span class="p">])</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># If this element was also at the root of row v, we need to remove the</span>
<span class="c1"># duplicate entry from H</span>
<span class="k">if</span> <span class="n">dq_heap</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">heap</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">v</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
<span class="n">H</span><span class="o">.</span><span class="n">remove</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="c1"># Remove best merge from row v heap</span>
<span class="n">dq_heap</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">remove</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="c1"># Push new row max onto H</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">v</span><span class="p">])</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">dq_heap</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># Duplicate wasn't in H, just remove from row v heap</span>
<span class="n">dq_heap</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">remove</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="c1"># Perform merge</span>
<span class="n">communities</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">communities</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">|</span> <span class="n">communities</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="k">del</span> <span class="n">communities</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="c1"># Get neighbor communities connected to the merged communities</span>
<span class="n">u_nbrs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">])</span>
<span class="n">v_nbrs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">dq_dict</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="n">all_nbrs</span> <span class="o">=</span> <span class="p">(</span><span class="n">u_nbrs</span> <span class="o">|</span> <span class="n">v_nbrs</span><span class="p">)</span> <span class="o">-</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">both_nbrs</span> <span class="o">=</span> <span class="n">u_nbrs</span> <span class="o">&</span> <span class="n">v_nbrs</span>
<span class="c1"># Update dq for merge of u into v</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">all_nbrs</span><span class="p">:</span>
<span class="c1"># Calculate new dq value</span>
<span class="k">if</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">both_nbrs</span><span class="p">:</span>
<span class="n">dq_vw</span> <span class="o">=</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">+</span> <span class="n">dq_dict</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="k">elif</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">v_nbrs</span><span class="p">:</span>
<span class="n">dq_vw</span> <span class="o">=</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">-</span> <span class="n">resolution</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">w</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="n">w</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">u</span><span class="p">])</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># w in u_nbrs</span>
<span class="n">dq_vw</span> <span class="o">=</span> <span class="n">dq_dict</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="o">-</span> <span class="n">resolution</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">w</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="n">w</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="c1"># Update rows v and w</span>
<span class="k">for</span> <span class="n">row</span><span class="p">,</span> <span class="n">col</span> <span class="ow">in</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="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">)]:</span>
<span class="n">dq_heap_row</span> <span class="o">=</span> <span class="n">dq_heap</span><span class="p">[</span><span class="n">row</span><span class="p">]</span>
<span class="c1"># Update dict for v,w only (u is removed below)</span>
<span class="n">dq_dict</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">dq_vw</span>
<span class="c1"># Save old max of per-row heap</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dq_heap_row</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">d_oldmax</span> <span class="o">=</span> <span class="n">dq_heap_row</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">d_oldmax</span> <span class="o">=</span> <span class="kc">None</span>
<span class="c1"># Add/update heaps</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">row</span><span class="p">,</span> <span class="n">col</span><span class="p">)</span>
<span class="n">d_negdq</span> <span class="o">=</span> <span class="o">-</span><span class="n">dq_vw</span>
<span class="c1"># Save old value for finding heap index</span>
<span class="k">if</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">v_nbrs</span><span class="p">:</span>
<span class="c1"># Update existing element in per-row heap</span>
<span class="n">dq_heap_row</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">priority</span><span class="o">=</span><span class="n">d_negdq</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># We're creating a new nonzero element, add to heap</span>
<span class="n">dq_heap_row</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">priority</span><span class="o">=</span><span class="n">d_negdq</span><span class="p">)</span>
<span class="c1"># Update heap of row maxes if necessary</span>
<span class="k">if</span> <span class="n">d_oldmax</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="c1"># No entries previously in this row, push new max</span>
<span class="n">H</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">priority</span><span class="o">=</span><span class="n">d_negdq</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># We've updated an entry in this row, has the max changed?</span>
<span class="n">row_max</span> <span class="o">=</span> <span class="n">dq_heap_row</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="n">d_oldmax</span> <span class="o">!=</span> <span class="n">row_max</span> <span class="ow">or</span> <span class="n">d_oldmax</span><span class="o">.</span><span class="n">priority</span> <span class="o">!=</span> <span class="n">row_max</span><span class="o">.</span><span class="n">priority</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">d_oldmax</span><span class="p">,</span> <span class="n">row_max</span><span class="p">)</span>
<span class="c1"># Remove row/col u from dq_dict matrix</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="c1"># Remove from dict</span>
<span class="n">dq_old</span> <span class="o">=</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">w</span><span class="p">][</span><span class="n">u</span><span class="p">]</span>
<span class="k">del</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">w</span><span class="p">][</span><span class="n">u</span><span class="p">]</span>
<span class="c1"># Remove from heaps if we haven't already</span>
<span class="k">if</span> <span class="n">w</span> <span class="o">!=</span> <span class="n">v</span><span class="p">:</span>
<span class="c1"># Remove both row and column</span>
<span class="k">for</span> <span class="n">row</span><span class="p">,</span> <span class="n">col</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">w</span><span class="p">,</span> <span class="n">u</span><span class="p">),</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="n">dq_heap_row</span> <span class="o">=</span> <span class="n">dq_heap</span><span class="p">[</span><span class="n">row</span><span class="p">]</span>
<span class="c1"># Check if replaced dq is row max</span>
<span class="n">d_old</span> <span class="o">=</span> <span class="p">(</span><span class="n">row</span><span class="p">,</span> <span class="n">col</span><span class="p">)</span>
<span class="k">if</span> <span class="n">dq_heap_row</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">d_old</span><span class="p">:</span>
<span class="c1"># Update per-row heap and heap of row maxes</span>
<span class="n">dq_heap_row</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d_old</span><span class="p">)</span>
<span class="n">H</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d_old</span><span class="p">)</span>
<span class="c1"># Update row max</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dq_heap_row</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">dq_heap_row</span><span class="o">.</span><span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># Only update per-row heap</span>
<span class="n">dq_heap_row</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d_old</span><span class="p">)</span>
<span class="k">del</span> <span class="n">dq_dict</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="c1"># Mark row u as deleted, but keep placeholder</span>
<span class="n">dq_heap</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="n">MappedQueue</span><span class="p">()</span>
<span class="c1"># Merge u into v and update a</span>
<span class="n">a</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+=</span> <span class="n">a</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">a</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="n">directed</span><span class="p">:</span>
<span class="n">b</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+=</span> <span class="n">b</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">b</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">yield</span> <span class="n">communities</span><span class="o">.</span><span class="n">values</span><span class="p">()</span>
<div class="viewcode-block" id="greedy_modularity_communities"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.community.modularity_max.greedy_modularity_communities.html#networkx.algorithms.community.modularity_max.greedy_modularity_communities">[docs]</a><span class="k">def</span> <span class="nf">greedy_modularity_communities</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">resolution</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">cutoff</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">best_n</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Find communities in G using greedy modularity maximization.</span>
<span class="sd"> This function uses Clauset-Newman-Moore greedy modularity maximization [2]_</span>
<span class="sd"> to find the community partition with the largest modularity.</span>
<span class="sd"> Greedy modularity maximization begins with each node in its own community</span>
<span class="sd"> and repeatedly joins the pair of communities that lead to the largest</span>
<span class="sd"> modularity until no futher increase in modularity is possible (a maximum).</span>
<span class="sd"> Two keyword arguments adjust the stopping condition. `cutoff` is a lower</span>
<span class="sd"> limit on the number of communities so you can stop the process before</span>
<span class="sd"> reaching a maximum (used to save computation time). `best_n` is an upper</span>
<span class="sd"> limit on the number of communities so you can make the process continue</span>
<span class="sd"> until at most n communities remain even if the maximum modularity occurs</span>
<span class="sd"> for more. To obtain exactly n communities, set both `cutoff` and `best_n` to n.</span>
<span class="sd"> This function maximizes the generalized modularity, where `resolution`</span>
<span class="sd"> is the resolution parameter, often expressed as $\gamma$.</span>
<span class="sd"> See :func:`~networkx.algorithms.community.quality.modularity`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or None, optional (default=None)</span>
<span class="sd"> The name of an edge attribute that holds the numerical value used</span>
<span class="sd"> as a weight. If None, then each edge has weight 1.</span>
<span class="sd"> The degree is the sum of the edge weights adjacent to the node.</span>
<span class="sd"> resolution : float, optional (default=1)</span>
<span class="sd"> If resolution is less than 1, modularity favors larger communities.</span>
<span class="sd"> Greater than 1 favors smaller communities.</span>
<span class="sd"> cutoff : int, optional (default=1)</span>
<span class="sd"> A minimum number of communities below which the merging process stops.</span>
<span class="sd"> The process stops at this number of communities even if modularity</span>
<span class="sd"> is not maximized. The goal is to let the user stop the process early.</span>
<span class="sd"> The process stops before the cutoff if it finds a maximum of modularity.</span>
<span class="sd"> best_n : int or None, optional (default=None)</span>
<span class="sd"> A maximum number of communities above which the merging process will</span>
<span class="sd"> not stop. This forces community merging to continue after modularity</span>
<span class="sd"> starts to decrease until `best_n` communities remain.</span>
<span class="sd"> If ``None``, don't force it to continue beyond a maximum.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ValueError : If the `cutoff` or `best_n` value is not in the range</span>
<span class="sd"> ``[1, G.number_of_nodes()]``, or if `best_n` < `cutoff`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> communities: list</span>
<span class="sd"> A list of frozensets of nodes, one for each community.</span>
<span class="sd"> Sorted by length with largest communities first.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms.community import greedy_modularity_communities</span>
<span class="sd"> >>> G = nx.karate_club_graph()</span>
<span class="sd"> >>> c = greedy_modularity_communities(G)</span>
<span class="sd"> >>> sorted(c[0])</span>
<span class="sd"> [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> modularity</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Newman, M. E. J. "Networks: An Introduction", page 224</span>
<span class="sd"> Oxford University Press 2011.</span>
<span class="sd"> .. [2] Clauset, A., Newman, M. E., & Moore, C.</span>
<span class="sd"> "Finding community structure in very large networks."</span>
<span class="sd"> Physical Review E 70(6), 2004.</span>
<span class="sd"> .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community</span>
<span class="sd"> Detection" Phys. Rev. E74, 2006.</span>
<span class="sd"> .. [4] Newman, M. E. J."Analysis of weighted networks"</span>
<span class="sd"> Physical Review E 70(5 Pt 2):056131, 2004.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="p">(</span><span class="n">cutoff</span> <span class="o"><</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">cutoff</span> <span class="o">></span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</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">"cutoff must be between 1 and </span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span><span class="si">}</span><span class="s2">. Got </span><span class="si">{</span><span class="n">cutoff</span><span class="si">}</span><span class="s2">."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">best_n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">best_n</span> <span class="o"><</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">best_n</span> <span class="o">></span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</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">"best_n must be between 1 and </span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span><span class="si">}</span><span class="s2">. Got </span><span class="si">{</span><span class="n">best_n</span><span class="si">}</span><span class="s2">."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">best_n</span> <span class="o"><</span> <span class="n">cutoff</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">"Must have best_n >= cutoff. Got </span><span class="si">{</span><span class="n">best_n</span><span class="si">}</span><span class="s2"> < </span><span class="si">{</span><span class="n">cutoff</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">best_n</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="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">best_n</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span>
<span class="c1"># retrieve generator object to construct output</span>
<span class="n">community_gen</span> <span class="o">=</span> <span class="n">_greedy_modularity_communities_generator</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">,</span> <span class="n">resolution</span><span class="o">=</span><span class="n">resolution</span>
<span class="p">)</span>
<span class="c1"># construct the first best community</span>
<span class="n">communities</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">community_gen</span><span class="p">)</span>
<span class="c1"># continue merging communities until one of the breaking criteria is satisfied</span>
<span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">communities</span><span class="p">)</span> <span class="o">></span> <span class="n">cutoff</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">dq</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">community_gen</span><span class="p">)</span>
<span class="c1"># StopIteration occurs when communities are the connected components</span>
<span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
<span class="n">communities</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">communities</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># if best_n requires more merging, merge big sets for highest modularity</span>
<span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">communities</span><span class="p">)</span> <span class="o">></span> <span class="n">best_n</span><span class="p">:</span>
<span class="n">comm1</span><span class="p">,</span> <span class="n">comm2</span><span class="p">,</span> <span class="o">*</span><span class="n">rest</span> <span class="o">=</span> <span class="n">communities</span>
<span class="n">communities</span> <span class="o">=</span> <span class="p">[</span><span class="n">comm1</span> <span class="o">^</span> <span class="n">comm2</span><span class="p">]</span>
<span class="n">communities</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">rest</span><span class="p">)</span>
<span class="k">return</span> <span class="n">communities</span>
<span class="c1"># keep going unless max_mod is reached or best_n says to merge more</span>
<span class="k">if</span> <span class="n">dq</span> <span class="o"><</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">communities</span><span class="p">)</span> <span class="o"><=</span> <span class="n">best_n</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">communities</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">community_gen</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">communities</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span></div>
<div class="viewcode-block" id="naive_greedy_modularity_communities"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.community.modularity_max.naive_greedy_modularity_communities.html#networkx.algorithms.community.modularity_max.naive_greedy_modularity_communities">[docs]</a><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">naive_greedy_modularity_communities</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">resolution</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Find communities in G using greedy modularity maximization.</span>
<span class="sd"> This implementation is O(n^4), much slower than alternatives, but it is</span>
<span class="sd"> provided as an easy-to-understand reference implementation.</span>
<span class="sd"> Greedy modularity maximization begins with each node in its own community</span>
<span class="sd"> and joins the pair of communities that most increases modularity until no</span>
<span class="sd"> such pair exists.</span>
<span class="sd"> This function maximizes the generalized modularity, where `resolution`</span>
<span class="sd"> is the resolution parameter, often expressed as $\gamma$.</span>
<span class="sd"> See :func:`~networkx.algorithms.community.quality.modularity`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> resolution : float (default=1)</span>
<span class="sd"> If resolution is less than 1, modularity favors larger communities.</span>
<span class="sd"> Greater than 1 favors smaller communities.</span>
<span class="sd"> weight : string or None, optional (default=None)</span>
<span class="sd"> The name of an edge attribute that holds the numerical value used</span>
<span class="sd"> as a weight. If None, then each edge has weight 1.</span>
<span class="sd"> The degree is the sum of the edge weights adjacent to the node.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> A list of sets of nodes, one for each community.</span>
<span class="sd"> Sorted by length with largest communities first.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms.community import \</span>
<span class="sd"> ... naive_greedy_modularity_communities</span>
<span class="sd"> >>> G = nx.karate_club_graph()</span>
<span class="sd"> >>> c = naive_greedy_modularity_communities(G)</span>
<span class="sd"> >>> sorted(c[0])</span>
<span class="sd"> [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> greedy_modularity_communities</span>
<span class="sd"> modularity</span>
<span class="sd"> """</span>
<span class="c1"># First create one community for each node</span>
<span class="n">communities</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">frozenset</span><span class="p">([</span><span class="n">u</span><span class="p">])</span> <span class="k">for</span> <span class="n">u</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="c1"># Track merges</span>
<span class="n">merges</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># Greedily merge communities until no improvement is possible</span>
<span class="n">old_modularity</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">new_modularity</span> <span class="o">=</span> <span class="n">modularity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">communities</span><span class="p">,</span> <span class="n">resolution</span><span class="o">=</span><span class="n">resolution</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">while</span> <span class="n">old_modularity</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">new_modularity</span> <span class="o">></span> <span class="n">old_modularity</span><span class="p">:</span>
<span class="c1"># Save modularity for comparison</span>
<span class="n">old_modularity</span> <span class="o">=</span> <span class="n">new_modularity</span>
<span class="c1"># Find best pair to merge</span>
<span class="n">trial_communities</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">communities</span><span class="p">)</span>
<span class="n">to_merge</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">communities</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">communities</span><span class="p">):</span>
<span class="c1"># Skip i==j and empty communities</span>
<span class="k">if</span> <span class="n">j</span> <span class="o"><=</span> <span class="n">i</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">continue</span>
<span class="c1"># Merge communities u and v</span>
<span class="n">trial_communities</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span> <span class="o">|</span> <span class="n">v</span>
<span class="n">trial_communities</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">([])</span>
<span class="n">trial_modularity</span> <span class="o">=</span> <span class="n">modularity</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">trial_communities</span><span class="p">,</span> <span class="n">resolution</span><span class="o">=</span><span class="n">resolution</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">trial_modularity</span> <span class="o">>=</span> <span class="n">new_modularity</span><span class="p">:</span>
<span class="c1"># Check if strictly better or tie</span>
<span class="k">if</span> <span class="n">trial_modularity</span> <span class="o">></span> <span class="n">new_modularity</span><span class="p">:</span>
<span class="c1"># Found new best, save modularity and group indexes</span>
<span class="n">new_modularity</span> <span class="o">=</span> <span class="n">trial_modularity</span>
<span class="n">to_merge</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">new_modularity</span> <span class="o">-</span> <span class="n">old_modularity</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">to_merge</span> <span class="ow">and</span> <span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o"><</span> <span class="nb">min</span><span class="p">(</span><span class="n">to_merge</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">to_merge</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
<span class="c1"># Break ties by choosing pair with lowest min id</span>
<span class="n">new_modularity</span> <span class="o">=</span> <span class="n">trial_modularity</span>
<span class="n">to_merge</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">new_modularity</span> <span class="o">-</span> <span class="n">old_modularity</span><span class="p">)</span>
<span class="c1"># Un-merge</span>
<span class="n">trial_communities</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
<span class="n">trial_communities</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
<span class="k">if</span> <span class="n">to_merge</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="c1"># If the best merge improves modularity, use it</span>
<span class="n">merges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">to_merge</span><span class="p">)</span>
<span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">dq</span> <span class="o">=</span> <span class="n">to_merge</span>
<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">communities</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">communities</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="n">communities</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span> <span class="o">|</span> <span class="n">v</span>
<span class="n">communities</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">([])</span>
<span class="c1"># Remove empty communities and sort</span>
<span class="k">return</span> <span class="nb">sorted</span><span class="p">((</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">communities</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span></div>
</pre></div>
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