"""
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"]
[docs]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` is the number of nodes that can reach `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
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, 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() # create a reversed graph view
if distance is not None:
# use Dijkstra's algorithm with specified attribute as edge weight
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_centrality = {}
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
# normalize to number of nodes-1 in connected part
if wf_improved:
s = (len(sp) - 1.0) / (len_G - 1)
_closeness_centrality *= s
closeness_centrality[n] = _closeness_centrality
if u is not None:
return closeness_centrality[u]
else:
return closeness_centrality
[docs]@not_implemented_for("directed")
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")
# Unpack edge
(u, v) = edge
path_length = nx.single_source_shortest_path_length
if insertion:
# For edge insertion, we want shortest paths before the edge is inserted
du = path_length(G, u)
dv = path_length(G, v)
G.add_edge(u, v)
else:
G.remove_edge(u, v)
# For edge removal, we want shortest paths after the edge is removed
du = path_length(G, u)
dv = path_length(G, v)
if prev_cc is None:
return nx.closeness_centrality(G)
nodes = G.nodes()
closeness_centrality = {}
for n in nodes:
if n in du and n in dv and abs(du[n] - dv[n]) <= 1:
closeness_centrality[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
# normalize to number of nodes-1 in connected part
if wf_improved:
s = (len(sp) - 1.0) / (len_G - 1)
_closeness_centrality *= s
closeness_centrality[n] = _closeness_centrality
# Leave the graph as we found it
if insertion:
G.remove_edge(u, v)
else:
G.add_edge(u, v)
return closeness_centrality