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<h1>Source code for networkx.algorithms.centrality.closeness</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Closeness centrality measures.</span>
<span class="sd">"""</span>
<span class="kn">import</span> <span class="nn">functools</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.exception</span> <span class="kn">import</span> <span class="n">NetworkXError</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">"closeness_centrality"</span><span class="p">,</span> <span class="s2">"incremental_closeness_centrality"</span><span class="p">]</span>
<div class="viewcode-block" id="closeness_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.closeness_centrality.html#networkx.algorithms.centrality.closeness_centrality">[docs]</a><span class="k">def</span> <span class="nf">closeness_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">distance</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">wf_improved</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Compute closeness centrality for nodes.</span>
<span class="sd"> Closeness centrality [1]_ of a node `u` is the reciprocal of the</span>
<span class="sd"> average shortest path distance to `u` over all `n-1` reachable nodes.</span>
<span class="sd"> .. math::</span>
<span class="sd"> C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},</span>
<span class="sd"> where `d(v, u)` is the shortest-path distance between `v` and `u`,</span>
<span class="sd"> and `n-1` is the number of nodes reachable from `u`. Notice that the</span>
<span class="sd"> closeness distance function computes the incoming distance to `u`</span>
<span class="sd"> for directed graphs. To use outward distance, act on `G.reverse()`.</span>
<span class="sd"> Notice that higher values of closeness indicate higher centrality.</span>
<span class="sd"> Wasserman and Faust propose an improved formula for graphs with</span>
<span class="sd"> more than one connected component. The result is "a ratio of the</span>
<span class="sd"> fraction of actors in the group who are reachable, to the average</span>
<span class="sd"> distance" from the reachable actors [2]_. You might think this</span>
<span class="sd"> scale factor is inverted but it is not. As is, nodes from small</span>
<span class="sd"> components receive a smaller closeness value. Letting `N` denote</span>
<span class="sd"> the number of nodes in the graph,</span>
<span class="sd"> .. math::</span>
<span class="sd"> C_{WF}(u) = \frac{n-1}{N-1} \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},</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"> u : node, optional</span>
<span class="sd"> Return only the value for node u</span>
<span class="sd"> distance : edge attribute key, optional (default=None)</span>
<span class="sd"> Use the specified edge attribute as the edge distance in shortest</span>
<span class="sd"> path calculations. If `None` (the default) all edges have a distance of 1.</span>
<span class="sd"> Absent edge attributes are assigned a distance of 1. Note that no check</span>
<span class="sd"> is performed to ensure that edges have the provided attribute.</span>
<span class="sd"> wf_improved : bool, optional (default=True)</span>
<span class="sd"> If True, scale by the fraction of nodes reachable. This gives the</span>
<span class="sd"> Wasserman and Faust improved formula. For single component graphs</span>
<span class="sd"> it is the same as the original formula.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> nodes : dictionary</span>
<span class="sd"> Dictionary of nodes with closeness centrality as the value.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])</span>
<span class="sd"> >>> nx.closeness_centrality(G)</span>
<span class="sd"> {0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> betweenness_centrality, load_centrality, eigenvector_centrality,</span>
<span class="sd"> degree_centrality, incremental_closeness_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The closeness centrality is normalized to `(n-1)/(|G|-1)` where</span>
<span class="sd"> `n` is the number of nodes in the connected part of graph</span>
<span class="sd"> containing the node. If the graph is not completely connected,</span>
<span class="sd"> this algorithm computes the closeness centrality for each</span>
<span class="sd"> connected part separately scaled by that parts size.</span>
<span class="sd"> If the 'distance' keyword is set to an edge attribute key then the</span>
<span class="sd"> shortest-path length will be computed using Dijkstra's algorithm with</span>
<span class="sd"> that edge attribute as the edge weight.</span>
<span class="sd"> The closeness centrality uses *inward* distance to a node, not outward.</span>
<span class="sd"> If you want to use outword distances apply the function to `G.reverse()`</span>
<span class="sd"> In NetworkX 2.2 and earlier a bug caused Dijkstra's algorithm to use the</span>
<span class="sd"> outward distance rather than the inward distance. If you use a 'distance'</span>
<span class="sd"> keyword and a DiGraph, your results will change between v2.2 and v2.3.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Linton C. Freeman: Centrality in networks: I.</span>
<span class="sd"> Conceptual clarification. Social Networks 1:215-239, 1979.</span>
<span class="sd"> https://doi.org/10.1016/0378-8733(78)90021-7</span>
<span class="sd"> .. [2] pg. 201 of Wasserman, S. and Faust, K.,</span>
<span class="sd"> Social Network Analysis: Methods and Applications, 1994,</span>
<span class="sd"> Cambridge University Press.</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"># create a reversed graph view</span>
<span class="k">if</span> <span class="n">distance</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="c1"># use Dijkstra's algorithm with specified attribute as edge weight</span>
<span class="n">path_length</span> <span class="o">=</span> <span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span>
<span class="n">nx</span><span class="o">.</span><span class="n">single_source_dijkstra_path_length</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">distance</span>
<span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">path_length</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">single_source_shortest_path_length</span>
<span class="k">if</span> <span class="n">u</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">closeness_dict</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="n">sp</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">totsp</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">len_G</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">_closeness_centrality</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">if</span> <span class="n">totsp</span> <span class="o">></span> <span class="mf">0.0</span> <span class="ow">and</span> <span class="n">len_G</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="n">_closeness_centrality</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sp</span><span class="p">)</span> <span class="o">-</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">/</span> <span class="n">totsp</span>
<span class="c1"># normalize to number of nodes-1 in connected part</span>
<span class="k">if</span> <span class="n">wf_improved</span><span class="p">:</span>
<span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sp</span><span class="p">)</span> <span class="o">-</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">len_G</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">_closeness_centrality</span> <span class="o">*=</span> <span class="n">s</span>
<span class="n">closeness_dict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">_closeness_centrality</span>
<span class="k">if</span> <span class="n">u</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="n">closeness_dict</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">return</span> <span class="n">closeness_dict</span></div>
<div class="viewcode-block" id="incremental_closeness_centrality"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.centrality.incremental_closeness_centrality.html#networkx.algorithms.centrality.incremental_closeness_centrality">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">incremental_closeness_centrality</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">edge</span><span class="p">,</span> <span class="n">prev_cc</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">insertion</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">wf_improved</span><span class="o">=</span><span class="kc">True</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Incremental closeness centrality for nodes.</span>
<span class="sd"> Compute closeness centrality for nodes using level-based work filtering</span>
<span class="sd"> as described in Incremental Algorithms for Closeness Centrality by Sariyuce et al.</span>
<span class="sd"> Level-based work filtering detects unnecessary updates to the closeness</span>
<span class="sd"> centrality and filters them out.</span>
<span class="sd"> ---</span>
<span class="sd"> From "Incremental Algorithms for Closeness Centrality":</span>
<span class="sd"> Theorem 1: Let :math:`G = (V, E)` be a graph and u and v be two vertices in V</span>
<span class="sd"> such that there is no edge (u, v) in E. Let :math:`G' = (V, E \cup uv)`</span>
<span class="sd"> Then :math:`cc[s] = cc'[s]` if and only if :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`.</span>
<span class="sd"> Where :math:`dG(u, v)` denotes the length of the shortest path between</span>
<span class="sd"> two vertices u, v in a graph G, cc[s] is the closeness centrality for a</span>
<span class="sd"> vertex s in V, and cc'[s] is the closeness centrality for a</span>
<span class="sd"> vertex s in V, with the (u, v) edge added.</span>
<span class="sd"> ---</span>
<span class="sd"> We use Theorem 1 to filter out updates when adding or removing an edge.</span>
<span class="sd"> When adding an edge (u, v), we compute the shortest path lengths from all</span>
<span class="sd"> other nodes to u and to v before the node is added. When removing an edge,</span>
<span class="sd"> we compute the shortest path lengths after the edge is removed. Then we</span>
<span class="sd"> apply Theorem 1 to use previously computed closeness centrality for nodes</span>
<span class="sd"> where :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`. This works only for</span>
<span class="sd"> undirected, unweighted graphs; the distance argument is not supported.</span>
<span class="sd"> Closeness centrality [1]_ of a node `u` is the reciprocal of the</span>
<span class="sd"> sum of the shortest path distances from `u` to all `n-1` other nodes.</span>
<span class="sd"> Since the sum of distances depends on the number of nodes in the</span>
<span class="sd"> graph, closeness is normalized by the sum of minimum possible</span>
<span class="sd"> distances `n-1`.</span>
<span class="sd"> .. math::</span>
<span class="sd"> C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},</span>
<span class="sd"> where `d(v, u)` is the shortest-path distance between `v` and `u`,</span>
<span class="sd"> and `n` is the number of nodes in the graph.</span>
<span class="sd"> Notice that higher values of closeness indicate higher centrality.</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"> edge : tuple</span>
<span class="sd"> The modified edge (u, v) in the graph.</span>
<span class="sd"> prev_cc : dictionary</span>
<span class="sd"> The previous closeness centrality for all nodes in the graph.</span>
<span class="sd"> insertion : bool, optional</span>
<span class="sd"> If True (default) the edge was inserted, otherwise it was deleted from the graph.</span>
<span class="sd"> wf_improved : bool, optional (default=True)</span>
<span class="sd"> If True, scale by the fraction of nodes reachable. This gives the</span>
<span class="sd"> Wasserman and Faust improved formula. For single component graphs</span>
<span class="sd"> it is the same as the original formula.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> nodes : dictionary</span>
<span class="sd"> Dictionary of nodes with closeness centrality as the value.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> betweenness_centrality, load_centrality, eigenvector_centrality,</span>
<span class="sd"> degree_centrality, closeness_centrality</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The closeness centrality is normalized to `(n-1)/(|G|-1)` where</span>
<span class="sd"> `n` is the number of nodes in the connected part of graph</span>
<span class="sd"> containing the node. If the graph is not completely connected,</span>
<span class="sd"> this algorithm computes the closeness centrality for each</span>
<span class="sd"> connected part separately.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Freeman, L.C., 1979. Centrality in networks: I.</span>
<span class="sd"> Conceptual clarification. Social Networks 1, 215--239.</span>
<span class="sd"> https://doi.org/10.1016/0378-8733(78)90021-7</span>
<span class="sd"> .. [2] Sariyuce, A.E. ; Kaya, K. ; Saule, E. ; Catalyiirek, U.V. Incremental</span>
<span class="sd"> Algorithms for Closeness Centrality. 2013 IEEE International Conference on Big Data</span>
<span class="sd"> http://sariyuce.com/papers/bigdata13.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">prev_cc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="nb">set</span><span class="p">(</span><span class="n">prev_cc</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="o">!=</span> <span class="nb">set</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="k">raise</span> <span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"prev_cc and G do not have the same nodes"</span><span class="p">)</span>
<span class="c1"># Unpack edge</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">edge</span>
<span class="n">path_length</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">single_source_shortest_path_length</span>
<span class="k">if</span> <span class="n">insertion</span><span class="p">:</span>
<span class="c1"># For edge insertion, we want shortest paths before the edge is inserted</span>
<span class="n">du</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
<span class="n">dv</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</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="k">else</span><span class="p">:</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="c1"># For edge removal, we want shortest paths after the edge is removed</span>
<span class="n">du</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
<span class="n">dv</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">if</span> <span class="n">prev_cc</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">closeness_centrality</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">nodes</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="n">closeness_dict</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="k">if</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">du</span> <span class="ow">and</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">dv</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">du</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="n">dv</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">closeness_dict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">prev_cc</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">sp</span> <span class="o">=</span> <span class="n">path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">totsp</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">len_G</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">_closeness_centrality</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">if</span> <span class="n">totsp</span> <span class="o">></span> <span class="mf">0.0</span> <span class="ow">and</span> <span class="n">len_G</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="n">_closeness_centrality</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sp</span><span class="p">)</span> <span class="o">-</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">/</span> <span class="n">totsp</span>
<span class="c1"># normalize to number of nodes-1 in connected part</span>
<span class="k">if</span> <span class="n">wf_improved</span><span class="p">:</span>
<span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sp</span><span class="p">)</span> <span class="o">-</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">len_G</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">_closeness_centrality</span> <span class="o">*=</span> <span class="n">s</span>
<span class="n">closeness_dict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">_closeness_centrality</span>
<span class="c1"># Leave the graph as we found it</span>
<span class="k">if</span> <span class="n">insertion</span><span class="p">:</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</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="k">return</span> <span class="n">closeness_dict</span></div>
</pre></div>
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