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<h1>Source code for networkx.algorithms.centrality.group</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Group centrality measures."""</span>
<span class="kn">from</span> <span class="nn">copy</span> <span class="kn">import</span> <span class="n">deepcopy</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.centrality.betweenness</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">_accumulate_endpoints</span><span class="p">,</span>
<span class="n">_single_source_dijkstra_path_basic</span><span class="p">,</span>
<span class="n">_single_source_shortest_path_basic</span><span class="p">,</span>
<span class="p">)</span>
<span class="kn">from</span> <span class="nn">networkx.utils.decorators</span> <span class="kn">import</span> <span class="n">not_implemented_for</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"group_betweenness_centrality"</span><span class="p">,</span>
<span class="s2">"group_closeness_centrality"</span><span class="p">,</span>
<span class="s2">"group_degree_centrality"</span><span class="p">,</span>
<span class="s2">"group_in_degree_centrality"</span><span class="p">,</span>
<span class="s2">"group_out_degree_centrality"</span><span class="p">,</span>
<span class="s2">"prominent_group"</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="group_betweenness_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.group_betweenness_centrality.html#networkx.algorithms.centrality.group_betweenness_centrality">[docs]</a><span class="k">def</span> <span class="nf">group_betweenness_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">normalized</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">endpoints</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Compute the group betweenness centrality for a group of nodes.</span>
<span class="sd"> Group betweenness centrality of a group of nodes $C$ is the sum of the</span>
<span class="sd"> fraction of all-pairs shortest paths that pass through any vertex in $C$</span>
<span class="sd"> .. math::</span>
<span class="sd"> c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}</span>
<span class="sd"> where $V$ is the set of nodes, $\sigma(s, t)$ is the number of</span>
<span class="sd"> shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of</span>
<span class="sd"> those paths passing through some node in group $C$. Note that</span>
<span class="sd"> $(s, t)$ are not members of the group ($V-C$ is the set of nodes</span>
<span class="sd"> in $V$ that are not in $C$).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> C : list or set or list of lists or list of sets</span>
<span class="sd"> A group or a list of groups containing nodes which belong to G, for which group betweenness</span>
<span class="sd"> centrality is to be calculated.</span>
<span class="sd"> normalized : bool, optional (default=True)</span>
<span class="sd"> If True, group betweenness is normalized by `1/((|V|-|C|)(|V|-|C|-1))`</span>
<span class="sd"> where `|V|` is the number of nodes in G and `|C|` is the number of nodes in C.</span>
<span class="sd"> weight : None or string, optional (default=None)</span>
<span class="sd"> If None, all edge weights are considered equal.</span>
<span class="sd"> Otherwise holds the name of the edge attribute used as weight.</span>
<span class="sd"> The weight of an edge is treated as the length or distance between the two sides.</span>
<span class="sd"> endpoints : bool, optional (default=False)</span>
<span class="sd"> If True include the endpoints in the shortest path counts.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If node(s) in C are not present in G.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> betweenness : list of floats or float</span>
<span class="sd"> If C is a single group then return a float. If C is a list with</span>
<span class="sd"> several groups then return a list of group betweenness centralities.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> betweenness_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.</span>
<span class="sd"> The initial implementation of the algorithm is mentioned in [2]_. This function uses</span>
<span class="sd"> an improved algorithm presented in [4]_.</span>
<span class="sd"> The number of nodes in the group must be a maximum of n - 2 where `n`</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> For weighted graphs the edge weights must be greater than zero.</span>
<span class="sd"> Zero edge weights can produce an infinite number of equal length</span>
<span class="sd"> paths between pairs of nodes.</span>
<span class="sd"> The total number of paths between source and target is counted</span>
<span class="sd"> differently for directed and undirected graphs. Directed paths</span>
<span class="sd"> between "u" and "v" are counted as two possible paths (one each</span>
<span class="sd"> direction) while undirected paths between "u" and "v" are counted</span>
<span class="sd"> as one path. Said another way, the sum in the expression above is</span>
<span class="sd"> over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] M G Everett and S P Borgatti:</span>
<span class="sd"> The Centrality of Groups and Classes.</span>
<span class="sd"> Journal of Mathematical Sociology. 23(3): 181-201. 1999.</span>
<span class="sd"> http://www.analytictech.com/borgatti/group_centrality.htm</span>
<span class="sd"> .. [2] Ulrik Brandes:</span>
<span class="sd"> On Variants of Shortest-Path Betweenness</span>
<span class="sd"> Centrality and their Generic Computation.</span>
<span class="sd"> Social Networks 30(2):136-145, 2008.</span>
<span class="sd"> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf</span>
<span class="sd"> .. [3] Sourav Medya et. al.:</span>
<span class="sd"> Group Centrality Maximization via Network Design.</span>
<span class="sd"> SIAM International Conference on Data Mining, SDM 2018, 126–134.</span>
<span class="sd"> https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf</span>
<span class="sd"> .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.</span>
<span class="sd"> "Fast algorithm for successive computation of group betweenness centrality."</span>
<span class="sd"> https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709</span>
<span class="sd"> """</span>
<span class="n">GBC</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># initialize betweenness</span>
<span class="n">list_of_groups</span> <span class="o">=</span> <span class="kc">True</span>
<span class="c1"># check weather C contains one or many groups</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">el</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">C</span><span class="p">):</span>
<span class="n">C</span> <span class="o">=</span> <span class="p">[</span><span class="n">C</span><span class="p">]</span>
<span class="n">list_of_groups</span> <span class="o">=</span> <span class="kc">False</span>
<span class="n">set_v</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span> <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">C</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">group</span><span class="p">}</span>
<span class="k">if</span> <span class="n">set_v</span> <span class="o">-</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">:</span> <span class="c1"># element(s) of C not in G</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The node(s) </span><span class="si">{</span><span class="n">set_v</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="si">}</span><span class="s2"> are in C but not in G."</span><span class="p">)</span>
<span class="c1"># pre-processing</span>
<span class="n">PB</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">_group_preprocessing</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">set_v</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="c1"># the algorithm for each group</span>
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">C</span><span class="p">:</span>
<span class="n">group</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">group</span><span class="p">)</span> <span class="c1"># set of nodes in group</span>
<span class="c1"># initialize the matrices of the sigma and the PB</span>
<span class="n">GBC_group</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">sigma_m</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
<span class="n">PB_m</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">PB</span><span class="p">)</span>
<span class="n">sigma_m_v</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">sigma_m</span><span class="p">)</span>
<span class="n">PB_m_v</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">PB_m</span><span class="p">)</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
<span class="n">GBC_group</span> <span class="o">+=</span> <span class="n">PB_m</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">v</span><span class="p">]</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
<span class="n">dxvy</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dxyv</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dvxy</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="ow">not</span> <span class="p">(</span>
<span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</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="ow">or</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span>
<span class="p">):</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="n">v</span><span class="p">]:</span>
<span class="n">dxyv</span> <span class="o">=</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">/</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">v</span><span class="p">]</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]:</span>
<span class="n">dxvy</span> <span class="o">=</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">*</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">/</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">x</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]:</span>
<span class="n">dvxy</span> <span class="o">=</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">x</span><span class="p">]</span> <span class="o">*</span> <span class="n">sigma</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">/</span> <span class="n">sigma</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="n">sigma_m_v</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigma_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">dxvy</span><span class="p">)</span>
<span class="n">PB_m_v</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="n">PB_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-</span> <span class="n">PB_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="n">dxvy</span>
<span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="n">v</span><span class="p">:</span>
<span class="n">PB_m_v</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-=</span> <span class="n">PB_m</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">*</span> <span class="n">dxyv</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">v</span><span class="p">:</span>
<span class="n">PB_m_v</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-=</span> <span class="n">PB_m</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="n">dvxy</span>
<span class="n">sigma_m</span><span class="p">,</span> <span class="n">sigma_m_v</span> <span class="o">=</span> <span class="n">sigma_m_v</span><span class="p">,</span> <span class="n">sigma_m</span>
<span class="n">PB_m</span><span class="p">,</span> <span class="n">PB_m_v</span> <span class="o">=</span> <span class="n">PB_m_v</span><span class="p">,</span> <span class="n">PB_m</span>
<span class="c1"># endpoints</span>
<span class="n">v</span><span class="p">,</span> <span class="n">c</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="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">endpoints</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># if the graph is connected then subtract the endpoints from</span>
<span class="c1"># the count for all the nodes in the graph. else count how many</span>
<span class="c1"># nodes are connected to the group's nodes and subtract that.</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_directed</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_strongly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">scale</span> <span class="o">=</span> <span class="n">c</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">v</span> <span class="o">-</span> <span class="n">c</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">scale</span> <span class="o">=</span> <span class="n">c</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">v</span> <span class="o">-</span> <span class="n">c</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="n">scale</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">for</span> <span class="n">group_node1</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">D</span><span class="p">[</span><span class="n">group_node1</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">node</span> <span class="o">!=</span> <span class="n">group_node1</span><span class="p">:</span>
<span class="k">if</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">+=</span> <span class="mi">2</span>
<span class="n">GBC_group</span> <span class="o">-=</span> <span class="n">scale</span>
<span class="c1"># normalized</span>
<span class="k">if</span> <span class="n">normalized</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span> <span class="o">-</span> <span class="n">c</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">GBC_group</span> <span class="o">*=</span> <span class="n">scale</span>
<span class="c1"># If undirected than count only the undirected edges</span>
<span class="k">elif</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">GBC_group</span> <span class="o">/=</span> <span class="mi">2</span>
<span class="n">GBC</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">GBC_group</span><span class="p">)</span>
<span class="k">if</span> <span class="n">list_of_groups</span><span class="p">:</span>
<span class="k">return</span> <span class="n">GBC</span>
<span class="k">return</span> <span class="n">GBC</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></div>
<span class="k">def</span> <span class="nf">_group_preprocessing</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">set_v</span><span class="p">,</span> <span class="n">weight</span><span class="p">):</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">delta</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">D</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">betweenness</span> <span class="o">=</span> <span class="nb">dict</span><span class="o">.</span><span class="n">fromkeys</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span> <span class="c1"># use BFS</span>
<span class="n">S</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">sigma</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">D</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">_single_source_shortest_path_basic</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># use Dijkstra's algorithm</span>
<span class="n">S</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">sigma</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">D</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">_single_source_dijkstra_path_basic</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">betweenness</span><span class="p">,</span> <span class="n">delta</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">_accumulate_endpoints</span><span class="p">(</span><span class="n">betweenness</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">sigma</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">s</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">delta</span><span class="p">[</span><span class="n">s</span><span class="p">]:</span> <span class="c1"># add the paths from s to i and rescale sigma</span>
<span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
<span class="n">delta</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">weight</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">sigma</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sigma</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">/</span> <span class="mi">2</span>
<span class="c1"># building the path betweenness matrix only for nodes that appear in the group</span>
<span class="n">PB</span> <span class="o">=</span> <span class="nb">dict</span><span class="o">.</span><span class="n">fromkeys</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">for</span> <span class="n">group_node1</span> <span class="ow">in</span> <span class="n">set_v</span><span class="p">:</span>
<span class="n">PB</span><span class="p">[</span><span class="n">group_node1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="o">.</span><span class="n">fromkeys</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">)</span>
<span class="k">for</span> <span class="n">group_node2</span> <span class="ow">in</span> <span class="n">set_v</span><span class="p">:</span>
<span class="k">if</span> <span class="n">group_node2</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">D</span><span class="p">[</span><span class="n">group_node1</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="c1"># if node is connected to the two group nodes than continue</span>
<span class="k">if</span> <span class="n">group_node2</span> <span class="ow">in</span> <span class="n">D</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="ow">and</span> <span class="n">group_node1</span> <span class="ow">in</span> <span class="n">D</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<span class="k">if</span> <span class="p">(</span>
<span class="n">D</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span>
<span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">group_node1</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">group_node1</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span>
<span class="p">):</span>
<span class="n">PB</span><span class="p">[</span><span class="n">group_node1</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span>
<span class="n">delta</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span>
<span class="o">*</span> <span class="n">sigma</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">group_node1</span><span class="p">]</span>
<span class="o">*</span> <span class="n">sigma</span><span class="p">[</span><span class="n">group_node1</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span>
<span class="o">/</span> <span class="n">sigma</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">group_node2</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">PB</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">D</span>
<div class="viewcode-block" id="prominent_group"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.prominent_group.html#networkx.algorithms.centrality.prominent_group">[docs]</a><span class="k">def</span> <span class="nf">prominent_group</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</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">C</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">endpoints</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">normalized</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="kc">False</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Find the prominent group of size $k$ in graph $G$. The prominence of the</span>
<span class="sd"> group is evaluated by the group betweenness centrality.</span>
<span class="sd"> Group betweenness centrality of a group of nodes $C$ is the sum of the</span>
<span class="sd"> fraction of all-pairs shortest paths that pass through any vertex in $C$</span>
<span class="sd"> .. math::</span>
<span class="sd"> c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}</span>
<span class="sd"> where $V$ is the set of nodes, $\sigma(s, t)$ is the number of</span>
<span class="sd"> shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of</span>
<span class="sd"> those paths passing through some node in group $C$. Note that</span>
<span class="sd"> $(s, t)$ are not members of the group ($V-C$ is the set of nodes</span>
<span class="sd"> in $V$ that are not in $C$).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> k : int</span>
<span class="sd"> The number of nodes in the group.</span>
<span class="sd"> normalized : bool, optional (default=True)</span>
<span class="sd"> If True, group betweenness is normalized by ``1/((|V|-|C|)(|V|-|C|-1))``</span>
<span class="sd"> where ``|V|`` is the number of nodes in G and ``|C|`` is the number of</span>
<span class="sd"> nodes in C.</span>
<span class="sd"> weight : None or string, optional (default=None)</span>
<span class="sd"> If None, all edge weights are considered equal.</span>
<span class="sd"> Otherwise holds the name of the edge attribute used as weight.</span>
<span class="sd"> The weight of an edge is treated as the length or distance between the two sides.</span>
<span class="sd"> endpoints : bool, optional (default=False)</span>
<span class="sd"> If True include the endpoints in the shortest path counts.</span>
<span class="sd"> C : list or set, optional (default=None)</span>
<span class="sd"> list of nodes which won't be candidates of the prominent group.</span>
<span class="sd"> greedy : bool, optional (default=False)</span>
<span class="sd"> Using a naive greedy algorithm in order to find non-optimal prominent</span>
<span class="sd"> group. For scale free networks the results are negligibly below the optimal</span>
<span class="sd"> results.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If node(s) in C are not present in G.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> max_GBC : float</span>
<span class="sd"> The group betweenness centrality of the prominent group.</span>
<span class="sd"> max_group : list</span>
<span class="sd"> The list of nodes in the prominent group.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> betweenness_centrality, group_betweenness_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.</span>
<span class="sd"> The algorithm is described in [2]_ and is based on techniques mentioned in [4]_.</span>
<span class="sd"> The number of nodes in the group must be a maximum of ``n - 2`` where ``n``</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> For weighted graphs the edge weights must be greater than zero.</span>
<span class="sd"> Zero edge weights can produce an infinite number of equal length</span>
<span class="sd"> paths between pairs of nodes.</span>
<span class="sd"> The total number of paths between source and target is counted</span>
<span class="sd"> differently for directed and undirected graphs. Directed paths</span>
<span class="sd"> between "u" and "v" are counted as two possible paths (one each</span>
<span class="sd"> direction) while undirected paths between "u" and "v" are counted</span>
<span class="sd"> as one path. Said another way, the sum in the expression above is</span>
<span class="sd"> over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] M G Everett and S P Borgatti:</span>
<span class="sd"> The Centrality of Groups and Classes.</span>
<span class="sd"> Journal of Mathematical Sociology. 23(3): 181-201. 1999.</span>
<span class="sd"> http://www.analytictech.com/borgatti/group_centrality.htm</span>
<span class="sd"> .. [2] Rami Puzis, Yuval Elovici, and Shlomi Dolev:</span>
<span class="sd"> "Finding the Most Prominent Group in Complex Networks"</span>
<span class="sd"> AI communications 20(4): 287-296, 2007.</span>
<span class="sd"> https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855</span>
<span class="sd"> .. [3] Sourav Medya et. al.:</span>
<span class="sd"> Group Centrality Maximization via Network Design.</span>
<span class="sd"> SIAM International Conference on Data Mining, SDM 2018, 126–134.</span>
<span class="sd"> https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf</span>
<span class="sd"> .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.</span>
<span class="sd"> "Fast algorithm for successive computation of group betweenness centrality."</span>
<span class="sd"> https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709</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">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="k">if</span> <span class="n">C</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">C</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
<span class="k">if</span> <span class="n">C</span> <span class="o">-</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">:</span> <span class="c1"># element(s) of C not in G</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The node(s) </span><span class="si">{</span><span class="n">C</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="si">}</span><span class="s2"> are in C but not in G."</span><span class="p">)</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span> <span class="o">-</span> <span class="n">C</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">)</span>
<span class="n">DF_tree</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="n">PB</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">_group_preprocessing</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">betweenness</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">PB</span><span class="p">)</span>
<span class="k">if</span> <span class="n">C</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">C</span><span class="p">:</span>
<span class="c1"># remove from the betweenness all the nodes not part of the group</span>
<span class="n">betweenness</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="n">index</span><span class="o">=</span><span class="n">node</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">betweenness</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="n">columns</span><span class="o">=</span><span class="n">node</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">CL</span> <span class="o">=</span> <span class="p">[</span><span class="n">node</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">node</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">betweenness</span><span class="p">),</span> <span class="n">nodes</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="n">max_GBC</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">max_group</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span>
<span class="mi">1</span><span class="p">,</span>
<span class="n">CL</span><span class="o">=</span><span class="n">CL</span><span class="p">,</span>
<span class="n">betweenness</span><span class="o">=</span><span class="n">betweenness</span><span class="p">,</span>
<span class="n">GBC</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">GM</span><span class="o">=</span><span class="p">[],</span>
<span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span>
<span class="n">cont</span><span class="o">=</span><span class="nb">dict</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">nodes</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">betweenness</span><span class="p">))),</span>
<span class="p">)</span>
<span class="c1"># the algorithm</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</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">k</span><span class="p">):</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span> <span class="o">+=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">][</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">][</span><span class="n">i</span><span class="p">]]</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="n">v</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="k">if</span> <span class="ow">not</span> <span class="n">endpoints</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># if the graph is connected then subtract the endpoints from</span>
<span class="c1"># the count for all the nodes in the graph. else count how many</span>
<span class="c1"># nodes are connected to the group's nodes and subtract that.</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_directed</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_strongly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">scale</span> <span class="o">=</span> <span class="n">k</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">v</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">scale</span> <span class="o">=</span> <span class="n">k</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">v</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="n">scale</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">for</span> <span class="n">group_node1</span> <span class="ow">in</span> <span class="n">max_group</span><span class="p">:</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">D</span><span class="p">[</span><span class="n">group_node1</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">node</span> <span class="o">!=</span> <span class="n">group_node1</span><span class="p">:</span>
<span class="k">if</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">max_group</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">+=</span> <span class="mi">2</span>
<span class="n">max_GBC</span> <span class="o">-=</span> <span class="n">scale</span>
<span class="c1"># normalized</span>
<span class="k">if</span> <span class="n">normalized</span><span class="p">:</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">max_GBC</span> <span class="o">*=</span> <span class="n">scale</span>
<span class="c1"># If undirected then count only the undirected edges</span>
<span class="k">elif</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">max_GBC</span> <span class="o">/=</span> <span class="mi">2</span>
<span class="n">max_GBC</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s2">"</span><span class="si">%.2f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">max_GBC</span><span class="p">)</span>
<span class="k">return</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">max_group</span></div>
<span class="k">def</span> <span class="nf">_dfbnb</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span><span class="p">):</span>
<span class="c1"># stopping condition - if we found a group of size k and with higher GBC then prune</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">])</span> <span class="o">==</span> <span class="n">k</span> <span class="ow">and</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">]</span> <span class="o">></span> <span class="n">max_GBC</span><span class="p">:</span>
<span class="k">return</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">],</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">]</span>
<span class="c1"># stopping condition - if the size of group members equal to k or there are less than</span>
<span class="c1"># k - |GM| in the candidate list or the heuristic function plus the GBC is bellow the</span>
<span class="c1"># maximal GBC found then prune</span>
<span class="k">if</span> <span class="p">(</span>
<span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">])</span> <span class="o">==</span> <span class="n">k</span>
<span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">])</span> <span class="o"><=</span> <span class="n">k</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">])</span>
<span class="ow">or</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">]</span> <span class="o">+</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span> <span class="o"><=</span> <span class="n">max_GBC</span>
<span class="p">):</span>
<span class="k">return</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span>
<span class="c1"># finding the heuristic of both children</span>
<span class="n">node_p</span><span class="p">,</span> <span class="n">node_m</span><span class="p">,</span> <span class="n">DF_tree</span> <span class="o">=</span> <span class="n">_heuristic</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span><span class="p">)</span>
<span class="c1"># finding the child with the bigger heuristic + GBC and expand</span>
<span class="c1"># that node first if greedy then only expand the plus node</span>
<span class="k">if</span> <span class="n">greedy</span><span class="p">:</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">node_p</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">]</span> <span class="o">+</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span>
<span class="o">></span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">]</span> <span class="o">+</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span>
<span class="p">):</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">node_p</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">node_m</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">node_m</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span> <span class="o">=</span> <span class="n">_dfbnb</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">node_p</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">max_group</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">max_GBC</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">max_group</span>
<span class="k">def</span> <span class="nf">_heuristic</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">DF_tree</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">greedy</span><span class="p">):</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># This helper function add two nodes to DF_tree - one left son and the</span>
<span class="c1"># other right son, finds their heuristic, CL, GBC, and GM</span>
<span class="n">node_p</span> <span class="o">=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">node_m</span> <span class="o">=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">+</span> <span class="mi">2</span>
<span class="n">added_node</span> <span class="o">=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># adding the plus node</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">add_nodes_from</span><span class="p">([(</span><span class="n">node_p</span><span class="p">,</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">]))])</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">added_node</span><span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"GBC"</span><span class="p">]</span> <span class="o">+=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span>
<span class="n">root_node</span> <span class="o">=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">]</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="n">dxvy</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dxyv</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dvxy</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="ow">not</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span>
<span class="ow">or</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span>
<span class="ow">or</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span>
<span class="p">):</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">y</span><span class="p">][</span><span class="n">added_node</span><span class="p">]:</span>
<span class="n">dxyv</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="o">*</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">y</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span>
<span class="o">/</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]:</span>
<span class="n">dxvy</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span>
<span class="o">*</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="o">/</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">==</span> <span class="n">D</span><span class="p">[</span><span class="n">added_node</span><span class="p">][</span><span class="n">x</span><span class="p">]</span> <span class="o">+</span> <span class="n">D</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]:</span>
<span class="n">dvxy</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">added_node</span><span class="p">][</span><span class="n">x</span><span class="p">]</span>
<span class="o">*</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="o">/</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]</span>
<span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"sigma"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">dxvy</span><span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-</span> <span class="n">root_node</span><span class="p">[</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="n">dxvy</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="n">added_node</span><span class="p">:</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">added_node</span><span class="p">]</span> <span class="o">*</span> <span class="n">dxyv</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">added_node</span><span class="p">:</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">x</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">-=</span> <span class="p">(</span>
<span class="n">root_node</span><span class="p">[</span><span class="s2">"betweenness"</span><span class="p">][</span><span class="n">added_node</span><span class="p">][</span><span class="n">y</span><span class="p">]</span> <span class="o">*</span> <span class="n">dvxy</span>
<span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span>
<span class="n">node</span>
<span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">node</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span>
<span class="nb">zip</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"betweenness"</span><span class="p">]),</span> <span class="n">nodes</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="k">if</span> <span class="n">node</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">]</span>
<span class="p">]</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
<span class="nb">zip</span><span class="p">(</span><span class="n">nodes</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"betweenness"</span><span class="p">]))</span>
<span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</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">k</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">])):</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span> <span class="o">+=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">][</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_p</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
<span class="p">]</span>
<span class="c1"># adding the minus node - don't insert the first node in the CL to GM</span>
<span class="c1"># Insert minus node only if isn't greedy type algorithm</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">greedy</span><span class="p">:</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">add_nodes_from</span><span class="p">([(</span><span class="n">node_m</span><span class="p">,</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">root</span><span class="p">]))])</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">]</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">]</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">added_node</span><span class="p">)</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</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">k</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"GM"</span><span class="p">])):</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"heu"</span><span class="p">]</span> <span class="o">+=</span> <span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"cont"</span><span class="p">][</span>
<span class="n">DF_tree</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">node_m</span><span class="p">][</span><span class="s2">"CL"</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
<span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">node_m</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">return</span> <span class="n">node_p</span><span class="p">,</span> <span class="n">node_m</span><span class="p">,</span> <span class="n">DF_tree</span>
<div class="viewcode-block" id="group_closeness_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.group_closeness_centrality.html#networkx.algorithms.centrality.group_closeness_centrality">[docs]</a><span class="k">def</span> <span class="nf">group_closeness_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Compute the group closeness centrality for a group of nodes.</span>
<span class="sd"> Group closeness centrality of a group of nodes $S$ is a measure</span>
<span class="sd"> of how close the group is to the other nodes in the graph.</span>
<span class="sd"> .. math::</span>
<span class="sd"> c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}}</span>
<span class="sd"> d_{S, v} = min_{u \in S} (d_{u, v})</span>
<span class="sd"> where $V$ is the set of nodes, $d_{S, v}$ is the distance of</span>
<span class="sd"> the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes</span>
<span class="sd"> in $V$ that are not in $S$).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> S : list or set</span>
<span class="sd"> S is a group of nodes which belong to G, for which group closeness</span>
<span class="sd"> centrality is to be calculated.</span>
<span class="sd"> weight : None or string, optional (default=None)</span>
<span class="sd"> If None, all edge weights are considered equal.</span>
<span class="sd"> Otherwise holds the name of the edge attribute used as weight.</span>
<span class="sd"> The weight of an edge is treated as the length or distance between the two sides.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If node(s) in S are not present in G.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> closeness : float</span>
<span class="sd"> Group closeness centrality of the group S.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> closeness_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The measure was introduced in [1]_.</span>
<span class="sd"> The formula implemented here is described in [2]_.</span>
<span class="sd"> Higher values of closeness indicate greater centrality.</span>
<span class="sd"> It is assumed that 1 / 0 is 0 (required in the case of directed graphs,</span>
<span class="sd"> or when a shortest path length is 0).</span>
<span class="sd"> The number of nodes in the group must be a maximum of n - 1 where `n`</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> For directed graphs, the incoming distance is utilized here. To use the</span>
<span class="sd"> outward distance, act on `G.reverse()`.</span>
<span class="sd"> For weighted graphs the edge weights must be greater than zero.</span>
<span class="sd"> Zero edge weights can produce an infinite number of equal length</span>
<span class="sd"> paths between pairs of nodes.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] M G Everett and S P Borgatti:</span>
<span class="sd"> The Centrality of Groups and Classes.</span>
<span class="sd"> Journal of Mathematical Sociology. 23(3): 181-201. 1999.</span>
<span class="sd"> http://www.analytictech.com/borgatti/group_centrality.htm</span>
<span class="sd"> .. [2] J. Zhao et. al.:</span>
<span class="sd"> Measuring and Maximizing Group Closeness Centrality over</span>
<span class="sd"> Disk Resident Graphs.</span>
<span class="sd"> WWWConference Proceedings, 2014. 689-694.</span>
<span class="sd"> https://doi.org/10.1145/2567948.2579356</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span> <span class="c1"># reverse view</span>
<span class="n">closeness</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># initialize to 0</span>
<span class="n">V</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="c1"># set of nodes in G</span>
<span class="n">S</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">S</span><span class="p">)</span> <span class="c1"># set of nodes in group S</span>
<span class="n">V_S</span> <span class="o">=</span> <span class="n">V</span> <span class="o">-</span> <span class="n">S</span> <span class="c1"># set of nodes in V but not S</span>
<span class="n">shortest_path_lengths</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">multi_source_dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="c1"># accumulation</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">V_S</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">closeness</span> <span class="o">+=</span> <span class="n">shortest_path_lengths</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span> <span class="c1"># no path exists</span>
<span class="n">closeness</span> <span class="o">+=</span> <span class="mi">0</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">closeness</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">V_S</span><span class="p">)</span> <span class="o">/</span> <span class="n">closeness</span>
<span class="k">except</span> <span class="ne">ZeroDivisionError</span><span class="p">:</span> <span class="c1"># 1 / 0 assumed as 0</span>
<span class="n">closeness</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">closeness</span></div>
<div class="viewcode-block" id="group_degree_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.group_degree_centrality.html#networkx.algorithms.centrality.group_degree_centrality">[docs]</a><span class="k">def</span> <span class="nf">group_degree_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Compute the group degree centrality for a group of nodes.</span>
<span class="sd"> Group degree centrality of a group of nodes $S$ is the fraction</span>
<span class="sd"> of non-group members connected to group members.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> S : list or set</span>
<span class="sd"> S is a group of nodes which belong to G, for which group degree</span>
<span class="sd"> centrality is to be calculated.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If node(s) in S are not in G.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> centrality : float</span>
<span class="sd"> Group degree centrality of the group S.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> degree_centrality</span>
<span class="sd"> group_in_degree_centrality</span>
<span class="sd"> group_out_degree_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The measure was introduced in [1]_.</span>
<span class="sd"> The number of nodes in the group must be a maximum of n - 1 where `n`</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] M G Everett and S P Borgatti:</span>
<span class="sd"> The Centrality of Groups and Classes.</span>
<span class="sd"> Journal of Mathematical Sociology. 23(3): 181-201. 1999.</span>
<span class="sd"> http://www.analytictech.com/borgatti/group_centrality.htm</span>
<span class="sd"> """</span>
<span class="n">centrality</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">()</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</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">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">S</span><span class="p">])</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">S</span><span class="p">))</span>
<span class="n">centrality</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">())</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="k">return</span> <span class="n">centrality</span></div>
<div class="viewcode-block" id="group_in_degree_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.group_in_degree_centrality.html#networkx.algorithms.centrality.group_in_degree_centrality">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">group_in_degree_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Compute the group in-degree centrality for a group of nodes.</span>
<span class="sd"> Group in-degree centrality of a group of nodes $S$ is the fraction</span>
<span class="sd"> of non-group members connected to group members by incoming edges.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> S : list or set</span>
<span class="sd"> S is a group of nodes which belong to G, for which group in-degree</span>
<span class="sd"> centrality is to be calculated.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> centrality : float</span>
<span class="sd"> Group in-degree centrality of the group S.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If G is undirected.</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If node(s) in S are not in G.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> degree_centrality</span>
<span class="sd"> group_degree_centrality</span>
<span class="sd"> group_out_degree_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The number of nodes in the group must be a maximum of n - 1 where `n`</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,</span>
<span class="sd"> so for group in-degree centrality, the reverse graph is used.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">group_degree_centrality</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">reverse</span><span class="p">(),</span> <span class="n">S</span><span class="p">)</span></div>
<div class="viewcode-block" id="group_out_degree_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.group_out_degree_centrality.html#networkx.algorithms.centrality.group_out_degree_centrality">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">group_out_degree_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Compute the group out-degree centrality for a group of nodes.</span>
<span class="sd"> Group out-degree centrality of a group of nodes $S$ is the fraction</span>
<span class="sd"> of non-group members connected to group members by outgoing edges.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A NetworkX graph.</span>
<span class="sd"> S : list or set</span>
<span class="sd"> S is a group of nodes which belong to G, for which group in-degree</span>
<span class="sd"> centrality is to be calculated.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> centrality : float</span>
<span class="sd"> Group out-degree centrality of the group S.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If G is undirected.</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If node(s) in S are not in G.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> degree_centrality</span>
<span class="sd"> group_degree_centrality</span>
<span class="sd"> group_in_degree_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The number of nodes in the group must be a maximum of n - 1 where `n`</span>
<span class="sd"> is the total number of nodes in the graph.</span>
<span class="sd"> `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,</span>
<span class="sd"> so for group out-degree centrality, the graph itself is used.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">group_degree_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span></div>
</pre></div>
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