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comparison env/lib/python3.9/site-packages/networkx/algorithms/centrality/closeness.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ | |
2 Closeness centrality measures. | |
3 """ | |
4 import functools | |
5 import networkx as nx | |
6 from networkx.exception import NetworkXError | |
7 from networkx.utils.decorators import not_implemented_for | |
8 | |
9 __all__ = ["closeness_centrality", "incremental_closeness_centrality"] | |
10 | |
11 | |
12 def closeness_centrality(G, u=None, distance=None, wf_improved=True): | |
13 r"""Compute closeness centrality for nodes. | |
14 | |
15 Closeness centrality [1]_ of a node `u` is the reciprocal of the | |
16 average shortest path distance to `u` over all `n-1` reachable nodes. | |
17 | |
18 .. math:: | |
19 | |
20 C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)}, | |
21 | |
22 where `d(v, u)` is the shortest-path distance between `v` and `u`, | |
23 and `n` is the number of nodes that can reach `u`. Notice that the | |
24 closeness distance function computes the incoming distance to `u` | |
25 for directed graphs. To use outward distance, act on `G.reverse()`. | |
26 | |
27 Notice that higher values of closeness indicate higher centrality. | |
28 | |
29 Wasserman and Faust propose an improved formula for graphs with | |
30 more than one connected component. The result is "a ratio of the | |
31 fraction of actors in the group who are reachable, to the average | |
32 distance" from the reachable actors [2]_. You might think this | |
33 scale factor is inverted but it is not. As is, nodes from small | |
34 components receive a smaller closeness value. Letting `N` denote | |
35 the number of nodes in the graph, | |
36 | |
37 .. math:: | |
38 | |
39 C_{WF}(u) = \frac{n-1}{N-1} \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)}, | |
40 | |
41 Parameters | |
42 ---------- | |
43 G : graph | |
44 A NetworkX graph | |
45 | |
46 u : node, optional | |
47 Return only the value for node u | |
48 | |
49 distance : edge attribute key, optional (default=None) | |
50 Use the specified edge attribute as the edge distance in shortest | |
51 path calculations | |
52 | |
53 wf_improved : bool, optional (default=True) | |
54 If True, scale by the fraction of nodes reachable. This gives the | |
55 Wasserman and Faust improved formula. For single component graphs | |
56 it is the same as the original formula. | |
57 | |
58 Returns | |
59 ------- | |
60 nodes : dictionary | |
61 Dictionary of nodes with closeness centrality as the value. | |
62 | |
63 See Also | |
64 -------- | |
65 betweenness_centrality, load_centrality, eigenvector_centrality, | |
66 degree_centrality, incremental_closeness_centrality | |
67 | |
68 Notes | |
69 ----- | |
70 The closeness centrality is normalized to `(n-1)/(|G|-1)` where | |
71 `n` is the number of nodes in the connected part of graph | |
72 containing the node. If the graph is not completely connected, | |
73 this algorithm computes the closeness centrality for each | |
74 connected part separately scaled by that parts size. | |
75 | |
76 If the 'distance' keyword is set to an edge attribute key then the | |
77 shortest-path length will be computed using Dijkstra's algorithm with | |
78 that edge attribute as the edge weight. | |
79 | |
80 The closeness centrality uses *inward* distance to a node, not outward. | |
81 If you want to use outword distances apply the function to `G.reverse()` | |
82 | |
83 In NetworkX 2.2 and earlier a bug caused Dijkstra's algorithm to use the | |
84 outward distance rather than the inward distance. If you use a 'distance' | |
85 keyword and a DiGraph, your results will change between v2.2 and v2.3. | |
86 | |
87 References | |
88 ---------- | |
89 .. [1] Linton C. Freeman: Centrality in networks: I. | |
90 Conceptual clarification. Social Networks 1:215-239, 1979. | |
91 http://leonidzhukov.ru/hse/2013/socialnetworks/papers/freeman79-centrality.pdf | |
92 .. [2] pg. 201 of Wasserman, S. and Faust, K., | |
93 Social Network Analysis: Methods and Applications, 1994, | |
94 Cambridge University Press. | |
95 """ | |
96 if G.is_directed(): | |
97 G = G.reverse() # create a reversed graph view | |
98 | |
99 if distance is not None: | |
100 # use Dijkstra's algorithm with specified attribute as edge weight | |
101 path_length = functools.partial( | |
102 nx.single_source_dijkstra_path_length, weight=distance | |
103 ) | |
104 else: | |
105 path_length = nx.single_source_shortest_path_length | |
106 | |
107 if u is None: | |
108 nodes = G.nodes | |
109 else: | |
110 nodes = [u] | |
111 closeness_centrality = {} | |
112 for n in nodes: | |
113 sp = path_length(G, n) | |
114 totsp = sum(sp.values()) | |
115 len_G = len(G) | |
116 _closeness_centrality = 0.0 | |
117 if totsp > 0.0 and len_G > 1: | |
118 _closeness_centrality = (len(sp) - 1.0) / totsp | |
119 # normalize to number of nodes-1 in connected part | |
120 if wf_improved: | |
121 s = (len(sp) - 1.0) / (len_G - 1) | |
122 _closeness_centrality *= s | |
123 closeness_centrality[n] = _closeness_centrality | |
124 if u is not None: | |
125 return closeness_centrality[u] | |
126 else: | |
127 return closeness_centrality | |
128 | |
129 | |
130 @not_implemented_for("directed") | |
131 def incremental_closeness_centrality( | |
132 G, edge, prev_cc=None, insertion=True, wf_improved=True | |
133 ): | |
134 r"""Incremental closeness centrality for nodes. | |
135 | |
136 Compute closeness centrality for nodes using level-based work filtering | |
137 as described in Incremental Algorithms for Closeness Centrality by Sariyuce et al. | |
138 | |
139 Level-based work filtering detects unnecessary updates to the closeness | |
140 centrality and filters them out. | |
141 | |
142 --- | |
143 From "Incremental Algorithms for Closeness Centrality": | |
144 | |
145 Theorem 1: Let :math:`G = (V, E)` be a graph and u and v be two vertices in V | |
146 such that there is no edge (u, v) in E. Let :math:`G' = (V, E \cup uv)` | |
147 Then :math:`cc[s] = cc'[s]` if and only if :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`. | |
148 | |
149 Where :math:`dG(u, v)` denotes the length of the shortest path between | |
150 two vertices u, v in a graph G, cc[s] is the closeness centrality for a | |
151 vertex s in V, and cc'[s] is the closeness centrality for a | |
152 vertex s in V, with the (u, v) edge added. | |
153 --- | |
154 | |
155 We use Theorem 1 to filter out updates when adding or removing an edge. | |
156 When adding an edge (u, v), we compute the shortest path lengths from all | |
157 other nodes to u and to v before the node is added. When removing an edge, | |
158 we compute the shortest path lengths after the edge is removed. Then we | |
159 apply Theorem 1 to use previously computed closeness centrality for nodes | |
160 where :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`. This works only for | |
161 undirected, unweighted graphs; the distance argument is not supported. | |
162 | |
163 Closeness centrality [1]_ of a node `u` is the reciprocal of the | |
164 sum of the shortest path distances from `u` to all `n-1` other nodes. | |
165 Since the sum of distances depends on the number of nodes in the | |
166 graph, closeness is normalized by the sum of minimum possible | |
167 distances `n-1`. | |
168 | |
169 .. math:: | |
170 | |
171 C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)}, | |
172 | |
173 where `d(v, u)` is the shortest-path distance between `v` and `u`, | |
174 and `n` is the number of nodes in the graph. | |
175 | |
176 Notice that higher values of closeness indicate higher centrality. | |
177 | |
178 Parameters | |
179 ---------- | |
180 G : graph | |
181 A NetworkX graph | |
182 | |
183 edge : tuple | |
184 The modified edge (u, v) in the graph. | |
185 | |
186 prev_cc : dictionary | |
187 The previous closeness centrality for all nodes in the graph. | |
188 | |
189 insertion : bool, optional | |
190 If True (default) the edge was inserted, otherwise it was deleted from the graph. | |
191 | |
192 wf_improved : bool, optional (default=True) | |
193 If True, scale by the fraction of nodes reachable. This gives the | |
194 Wasserman and Faust improved formula. For single component graphs | |
195 it is the same as the original formula. | |
196 | |
197 Returns | |
198 ------- | |
199 nodes : dictionary | |
200 Dictionary of nodes with closeness centrality as the value. | |
201 | |
202 See Also | |
203 -------- | |
204 betweenness_centrality, load_centrality, eigenvector_centrality, | |
205 degree_centrality, closeness_centrality | |
206 | |
207 Notes | |
208 ----- | |
209 The closeness centrality is normalized to `(n-1)/(|G|-1)` where | |
210 `n` is the number of nodes in the connected part of graph | |
211 containing the node. If the graph is not completely connected, | |
212 this algorithm computes the closeness centrality for each | |
213 connected part separately. | |
214 | |
215 References | |
216 ---------- | |
217 .. [1] Freeman, L.C., 1979. Centrality in networks: I. | |
218 Conceptual clarification. Social Networks 1, 215--239. | |
219 http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF | |
220 .. [2] Sariyuce, A.E. ; Kaya, K. ; Saule, E. ; Catalyiirek, U.V. Incremental | |
221 Algorithms for Closeness Centrality. 2013 IEEE International Conference on Big Data | |
222 http://sariyuce.com/papers/bigdata13.pdf | |
223 """ | |
224 if prev_cc is not None and set(prev_cc.keys()) != set(G.nodes()): | |
225 raise NetworkXError("prev_cc and G do not have the same nodes") | |
226 | |
227 # Unpack edge | |
228 (u, v) = edge | |
229 path_length = nx.single_source_shortest_path_length | |
230 | |
231 if insertion: | |
232 # For edge insertion, we want shortest paths before the edge is inserted | |
233 du = path_length(G, u) | |
234 dv = path_length(G, v) | |
235 | |
236 G.add_edge(u, v) | |
237 else: | |
238 G.remove_edge(u, v) | |
239 | |
240 # For edge removal, we want shortest paths after the edge is removed | |
241 du = path_length(G, u) | |
242 dv = path_length(G, v) | |
243 | |
244 if prev_cc is None: | |
245 return nx.closeness_centrality(G) | |
246 | |
247 nodes = G.nodes() | |
248 closeness_centrality = {} | |
249 for n in nodes: | |
250 if n in du and n in dv and abs(du[n] - dv[n]) <= 1: | |
251 closeness_centrality[n] = prev_cc[n] | |
252 else: | |
253 sp = path_length(G, n) | |
254 totsp = sum(sp.values()) | |
255 len_G = len(G) | |
256 _closeness_centrality = 0.0 | |
257 if totsp > 0.0 and len_G > 1: | |
258 _closeness_centrality = (len(sp) - 1.0) / totsp | |
259 # normalize to number of nodes-1 in connected part | |
260 if wf_improved: | |
261 s = (len(sp) - 1.0) / (len_G - 1) | |
262 _closeness_centrality *= s | |
263 closeness_centrality[n] = _closeness_centrality | |
264 | |
265 # Leave the graph as we found it | |
266 if insertion: | |
267 G.remove_edge(u, v) | |
268 else: | |
269 G.add_edge(u, v) | |
270 | |
271 return closeness_centrality |