"""Group centrality measures."""
from itertools import combinations
import networkx as nx
from networkx.utils.decorators import not_implemented_for
__all__ = [
"group_betweenness_centrality",
"group_closeness_centrality",
"group_degree_centrality",
"group_in_degree_centrality",
"group_out_degree_centrality",
]
[docs]def group_betweenness_centrality(G, C, normalized=True, weight=None):
r"""Compute the group betweenness centrality for a group of nodes.
Group betweenness centrality of a group of nodes $C$ is the sum of the
fraction of all-pairs shortest paths that pass through any vertex in $C$
.. math::
c_B(C) =\sum_{s,t \in V-C; s<t} \frac{\sigma(s, t|C)}{\sigma(s, t)}
where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of
those paths passing through some node in group $C$. Note that
$(s, t)$ are not members of the group ($V-C$ is the set of nodes
in $V$ that are not in $C$).
Parameters
----------
G : graph
A NetworkX graph.
C : list or set
C is a group of nodes which belong to G, for which group betweenness
centrality is to be calculated.
normalized : bool, optional
If True, group betweenness is normalized by `2/((|V|-|C|)(|V|-|C|-1))`
for graphs and `1/((|V|-|C|)(|V|-|C|-1))` for directed graphs where `|V|`
is the number of nodes in G and `|C|` is the number of nodes in C.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Raises
------
NodeNotFound
If node(s) in C are not present in G.
Returns
-------
betweenness : float
Group betweenness centrality of the group C.
See Also
--------
betweenness_centrality
Notes
-----
The measure is described in [1]_.
The algorithm is an extension of the one proposed by Ulrik Brandes for
betweenness centrality of nodes. Group betweenness is also mentioned in
his paper [2]_ along with the algorithm. The importance of the measure is
discussed in [3]_.
The number of nodes in the group must be a maximum of n - 2 where `n`
is the total number of nodes in the graph.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
References
----------
.. [1] M G Everett and S P Borgatti:
The Centrality of Groups and Classes.
Journal of Mathematical Sociology. 23(3): 181-201. 1999.
http://www.analytictech.com/borgatti/group_centrality.htm
.. [2] Ulrik Brandes:
On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf
.. [3] Sourav Medya et. al.:
Group Centrality Maximization via Network Design.
SIAM International Conference on Data Mining, SDM 2018, 126–134.
https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf
"""
betweenness = 0 # initialize betweenness to 0
V = set(G) # set of nodes in G
C = set(C) # set of nodes in C (group)
if len(C - V) != 0: # element(s) of C not in V
raise nx.NodeNotFound(
"The node(s) " + str(list(C - V)) + " are not " "in the graph."
)
V_C = V - C # set of nodes in V but not in C
# accumulation
for pair in combinations(V_C, 2): # (s, t) pairs of V_C
try:
paths = 0
paths_through_C = 0
for path in nx.all_shortest_paths(
G, source=pair[0], target=pair[1], weight=weight
):
if set(path) & C:
paths_through_C += 1
paths += 1
betweenness += paths_through_C / paths
except nx.exception.NetworkXNoPath:
betweenness += 0
# rescaling
v, c = len(G), len(C)
if normalized:
scale = 1 / ((v - c) * (v - c - 1))
if not G.is_directed():
scale *= 2
else:
scale = None
if scale is not None:
betweenness *= scale
return betweenness
[docs]def group_closeness_centrality(G, S, weight=None):
r"""Compute the group closeness centrality for a group of nodes.
Group closeness centrality of a group of nodes $S$ is a measure
of how close the group is to the other nodes in the graph.
.. math::
c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}}
d_{S, v} = min_{u \in S} (d_{u, v})
where $V$ is the set of nodes, $d_{S, v}$ is the distance of
the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes
in $V$ that are not in $S$).
Parameters
----------
G : graph
A NetworkX graph.
S : list or set
S is a group of nodes which belong to G, for which group closeness
centrality is to be calculated.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Raises
------
NodeNotFound
If node(s) in S are not present in G.
Returns
-------
closeness : float
Group closeness centrality of the group S.
See Also
--------
closeness_centrality
Notes
-----
The measure was introduced in [1]_.
The formula implemented here is described in [2]_.
Higher values of closeness indicate greater centrality.
It is assumed that 1 / 0 is 0 (required in the case of directed graphs,
or when a shortest path length is 0).
The number of nodes in the group must be a maximum of n - 1 where `n`
is the total number of nodes in the graph.
For directed graphs, the incoming distance is utilized here. To use the
outward distance, act on `G.reverse()`.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
References
----------
.. [1] M G Everett and S P Borgatti:
The Centrality of Groups and Classes.
Journal of Mathematical Sociology. 23(3): 181-201. 1999.
http://www.analytictech.com/borgatti/group_centrality.htm
.. [2] J. Zhao et. al.:
Measuring and Maximizing Group Closeness Centrality over
Disk Resident Graphs.
WWWConference Proceedings, 2014. 689-694.
http://wwwconference.org/proceedings/www2014/companion/p689.pdf
"""
if G.is_directed():
G = G.reverse() # reverse view
closeness = 0 # initialize to 0
V = set(G) # set of nodes in G
S = set(S) # set of nodes in group S
V_S = V - S # set of nodes in V but not S
shortest_path_lengths = nx.multi_source_dijkstra_path_length(G, S, weight=weight)
# accumulation
for v in V_S:
try:
closeness += shortest_path_lengths[v]
except KeyError: # no path exists
closeness += 0
try:
closeness = len(V_S) / closeness
except ZeroDivisionError: # 1 / 0 assumed as 0
closeness = 0
return closeness
[docs]def group_degree_centrality(G, S):
"""Compute the group degree centrality for a group of nodes.
Group degree centrality of a group of nodes $S$ is the fraction
of non-group members connected to group members.
Parameters
----------
G : graph
A NetworkX graph.
S : list or set
S is a group of nodes which belong to G, for which group degree
centrality is to be calculated.
Raises
------
NetworkXError
If node(s) in S are not in G.
Returns
-------
centrality : float
Group degree centrality of the group S.
See Also
--------
degree_centrality
group_in_degree_centrality
group_out_degree_centrality
Notes
-----
The measure was introduced in [1]_.
The number of nodes in the group must be a maximum of n - 1 where `n`
is the total number of nodes in the graph.
References
----------
.. [1] M G Everett and S P Borgatti:
The Centrality of Groups and Classes.
Journal of Mathematical Sociology. 23(3): 181-201. 1999.
http://www.analytictech.com/borgatti/group_centrality.htm
"""
centrality = len(set().union(*list(set(G.neighbors(i)) for i in S)) - set(S))
centrality /= len(G.nodes()) - len(S)
return centrality
[docs]@not_implemented_for("undirected")
def group_in_degree_centrality(G, S):
"""Compute the group in-degree centrality for a group of nodes.
Group in-degree centrality of a group of nodes $S$ is the fraction
of non-group members connected to group members by incoming edges.
Parameters
----------
G : graph
A NetworkX graph.
S : list or set
S is a group of nodes which belong to G, for which group in-degree
centrality is to be calculated.
Returns
-------
centrality : float
Group in-degree centrality of the group S.
Raises
------
NetworkXNotImplemented
If G is undirected.
NodeNotFound
If node(s) in S are not in G.
See Also
--------
degree_centrality
group_degree_centrality
group_out_degree_centrality
Notes
-----
The number of nodes in the group must be a maximum of n - 1 where `n`
is the total number of nodes in the graph.
`G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
so for group in-degree centrality, the reverse graph is used.
"""
return group_degree_centrality(G.reverse(), S)
[docs]@not_implemented_for("undirected")
def group_out_degree_centrality(G, S):
"""Compute the group out-degree centrality for a group of nodes.
Group out-degree centrality of a group of nodes $S$ is the fraction
of non-group members connected to group members by outgoing edges.
Parameters
----------
G : graph
A NetworkX graph.
S : list or set
S is a group of nodes which belong to G, for which group in-degree
centrality is to be calculated.
Returns
-------
centrality : float
Group out-degree centrality of the group S.
Raises
------
NetworkXNotImplemented
If G is undirected.
NodeNotFound
If node(s) in S are not in G.
See Also
--------
degree_centrality
group_degree_centrality
group_in_degree_centrality
Notes
-----
The number of nodes in the group must be a maximum of n - 1 where `n`
is the total number of nodes in the graph.
`G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
so for group out-degree centrality, the graph itself is used.
"""
return group_degree_centrality(G, S)