272 lines
9.7 KiB
Python
272 lines
9.7 KiB
Python
|
"""
|
||
|
|
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"]
|
||
|
|
|
||
|
|
|
||
|
|
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.
|
||
|
|
http://leonidzhukov.ru/hse/2013/socialnetworks/papers/freeman79-centrality.pdf
|
||
|
|
.. [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
|
||
|
|
|
||
|
|
|
||
|
|
@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.
|
||
|
|
http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF
|
||
|
|
.. [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
|