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