# Source code for networkx.algorithms.centrality.group

"""Group centrality measures."""
from copy import deepcopy

import networkx as nx
from networkx.utils.decorators import not_implemented_for
from networkx.algorithms.centrality.betweenness import (
_single_source_shortest_path_basic,
_single_source_dijkstra_path_basic,
_accumulate_endpoints,
)

__all__ = [
"group_betweenness_centrality",
"group_closeness_centrality",
"group_degree_centrality",
"group_in_degree_centrality",
"group_out_degree_centrality",
"prominent_group",
]

[docs]def group_betweenness_centrality(G, C, normalized=True, weight=None, endpoints=False):
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(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\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 or list of lists or list of sets
A group or a list of groups containing nodes which belong to G, for which group betweenness
centrality is to be calculated.

normalized : bool, optional (default=True)
If True, group betweenness is normalized by 1/((|V|-|C|)(|V|-|C|-1))
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.
The weight of an edge is treated as the length or distance between the two sides.

endpoints : bool, optional (default=False)
If True include the endpoints in the shortest path counts.

Raises
------
NodeNotFound
If node(s) in C are not present in G.

Returns
-------
betweenness : list of floats or float
If C is a single group then return a float. If C is a list with
several groups then return a list of group betweenness centralities.

See Also
--------
betweenness_centrality

Notes
-----
Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.
The initial implementation of the algorithm is mentioned in [2]_. This function uses
an improved algorithm presented in [4]_.

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.

The total number of paths between source and target is counted
differently for directed and undirected graphs. Directed paths
between "u" and "v" are counted as two possible paths (one each
direction) while undirected paths between "u" and "v" are counted
as one path. Said another way, the sum in the expression above is
over all s != t for directed graphs and for s < t for undirected graphs.

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
.. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.
"Fast algorithm for successive computation of group betweenness centrality."
https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709

"""
GBC = []  # initialize betweenness
list_of_groups = True
#  check weather C contains one or many groups
if any(el in G for el in C):
C = [C]
list_of_groups = False
set_v = {node for group in C for node in group}
if set_v - G.nodes:  # element(s) of C not in G
raise nx.NodeNotFound(f"The node(s) {set_v - G.nodes} are in C but not in G.")

# pre-processing
PB, sigma, D = _group_preprocessing(G, set_v, weight)

# the algorithm for each group
for group in C:
group = set(group)  # set of nodes in group
# initialize the matrices of the sigma and the PB
GBC_group = 0
sigma_m = deepcopy(sigma)
PB_m = deepcopy(PB)
sigma_m_v = deepcopy(sigma_m)
PB_m_v = deepcopy(PB_m)
for v in group:
GBC_group += PB_m[v][v]
for x in group:
for y in group:
dxvy = 0
dxyv = 0
dvxy = 0
if not (
sigma_m[x][y] == 0 or sigma_m[x][v] == 0 or sigma_m[v][y] == 0
):
if D[x][v] == D[x][y] + D[y][v]:
dxyv = sigma_m[x][y] * sigma_m[y][v] / sigma_m[x][v]
if D[x][y] == D[x][v] + D[v][y]:
dxvy = sigma_m[x][v] * sigma_m[v][y] / sigma_m[x][y]
if D[v][y] == D[v][x] + D[x][y]:
dvxy = sigma_m[v][x] * sigma[x][y] / sigma[v][y]
sigma_m_v[x][y] = sigma_m[x][y] * (1 - dxvy)
PB_m_v[x][y] = PB_m[x][y] - PB_m[x][y] * dxvy
if y != v:
PB_m_v[x][y] -= PB_m[x][v] * dxyv
if x != v:
PB_m_v[x][y] -= PB_m[v][y] * dvxy
sigma_m, sigma_m_v = sigma_m_v, sigma_m
PB_m, PB_m_v = PB_m_v, PB_m

# endpoints
v, c = len(G), len(group)
if not endpoints:
scale = 0
# if the graph is connected then subtract the endpoints from
# the count for all the nodes in the graph. else count how many
# nodes are connected to the group's nodes and subtract that.
if nx.is_directed(G):
if nx.is_strongly_connected(G):
scale = c * (2 * v - c - 1)
elif nx.is_connected(G):
scale = c * (2 * v - c - 1)
if scale == 0:
for group_node1 in group:
for node in D[group_node1]:
if node != group_node1:
if node in group:
scale += 1
else:
scale += 2
GBC_group -= scale

# normalized
if normalized:
scale = 1 / ((v - c) * (v - c - 1))
GBC_group *= scale

# If undirected than count only the undirected edges
elif not G.is_directed():
GBC_group /= 2

GBC.append(GBC_group)
if list_of_groups:
return GBC
else:
return GBC[0]

def _group_preprocessing(G, set_v, weight):
sigma = {}
delta = {}
D = {}
betweenness = dict.fromkeys(G, 0)
for s in G:
if weight is None:  # use BFS
S, P, sigma[s], D[s] = _single_source_shortest_path_basic(G, s)
else:  # use Dijkstra's algorithm
S, P, sigma[s], D[s] = _single_source_dijkstra_path_basic(G, s, weight)
betweenness, delta[s] = _accumulate_endpoints(betweenness, S, P, sigma[s], s)
for i in delta[s].keys():  # add the paths from s to i and rescale sigma
if s != i:
delta[s][i] += 1
if weight is not None:
sigma[s][i] = sigma[s][i] / 2
# building the path betweenness matrix only for nodes that appear in the group
PB = dict.fromkeys(G)
for group_node1 in set_v:
PB[group_node1] = dict.fromkeys(G, 0.0)
for group_node2 in set_v:
if group_node2 not in D[group_node1]:
continue
for node in G:
# if node is connected to the two group nodes than continue
if group_node2 in D[node] and group_node1 in D[node]:
if (
D[node][group_node2]
== D[node][group_node1] + D[group_node1][group_node2]
):
PB[group_node1][group_node2] += (
delta[node][group_node2]
* sigma[node][group_node1]
* sigma[group_node1][group_node2]
/ sigma[node][group_node2]
)
return PB, sigma, D

[docs]def prominent_group(
G, k, weight=None, C=None, endpoints=False, normalized=True, greedy=False
):
r"""Find the prominent group of size $k$ in graph $G$. The prominence of the
group is evaluated by the group betweenness centrality.

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(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\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.

k : int
The number of nodes in the group.

normalized : bool, optional (default=True)
If True, group betweenness is normalized by 1/((|V|-|C|)(|V|-|C|-1))
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.
The weight of an edge is treated as the length or distance between the two sides.

endpoints : bool, optional (default=False)
If True include the endpoints in the shortest path counts.

C : list or set, optional (default=None)
list of nodes which won't be candidates of the prominent group.

greedy : bool, optional (default=False)
Using a naive greedy algorithm in order to find non-optimal prominent
group. For scale free networks the results are negligibly below the optimal
results.

Raises
------
NodeNotFound
If node(s) in C are not present in G.

Returns
-------
max_GBC : float
The group betweenness centrality of the prominent group.

max_group : list
The list of nodes in the prominent group.

See Also
--------
betweenness_centrality, group_betweenness_centrality

Notes
-----
Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.
The algorithm is described in [2]_ and is based on techniques mentioned in [4]_.

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.

The total number of paths between source and target is counted
differently for directed and undirected graphs. Directed paths
between "u" and "v" are counted as two possible paths (one each
direction) while undirected paths between "u" and "v" are counted
as one path. Said another way, the sum in the expression above is
over all s != t for directed graphs and for s < t for undirected graphs.

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] Rami Puzis, Yuval Elovici, and Shlomi Dolev:
"Finding the Most Prominent Group in Complex Networks"
AI communications 20(4): 287-296, 2007.
https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855
.. [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
.. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.
"Fast algorithm for successive computation of group betweenness centrality."
https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709
"""
import pandas as pd
import numpy as np

if C is not None:
C = set(C)
if C - G.nodes:  # element(s) of C not in G
raise nx.NodeNotFound(f"The node(s) {C - G.nodes} are in C but not in G.")
nodes = list(G.nodes - C)
else:
nodes = list(G.nodes)
DF_tree = nx.Graph()
PB, sigma, D = _group_preprocessing(G, nodes, weight)
betweenness = pd.DataFrame.from_dict(PB)
if C is not None:
for node in C:
# remove from the betweenness all the nodes not part of the group
betweenness.drop(index=node, inplace=True)
betweenness.drop(columns=node, inplace=True)
CL = [node for _, node in sorted(zip(np.diag(betweenness), nodes), reverse=True)]
max_GBC = 0
max_group = []
DF_tree.add_node(
1,
CL=CL,
betweenness=betweenness,
GBC=0,
GM=[],
sigma=sigma,
cont=dict(zip(nodes, np.diag(betweenness))),
)

# the algorithm
DF_tree.nodes[1]["heu"] = 0
for i in range(k):
DF_tree.nodes[1]["heu"] += DF_tree.nodes[1]["cont"][DF_tree.nodes[1]["CL"][i]]
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, 1, D, max_group, nodes, greedy
)

v = len(G)
if not endpoints:
scale = 0
# if the graph is connected then subtract the endpoints from
# the count for all the nodes in the graph. else count how many
# nodes are connected to the group's nodes and subtract that.
if nx.is_directed(G):
if nx.is_strongly_connected(G):
scale = k * (2 * v - k - 1)
elif nx.is_connected(G):
scale = k * (2 * v - k - 1)
if scale == 0:
for group_node1 in max_group:
for node in D[group_node1]:
if node != group_node1:
if node in max_group:
scale += 1
else:
scale += 2
max_GBC -= scale

# normalized
if normalized:
scale = 1 / ((v - k) * (v - k - 1))
max_GBC *= scale

# If undirected then count only the undirected edges
elif not G.is_directed():
max_GBC /= 2
max_GBC = float("%.2f" % max_GBC)
return max_GBC, max_group

def _dfbnb(G, k, DF_tree, max_GBC, root, D, max_group, nodes, greedy):
# stopping condition - if we found a group of size k and with higher GBC then prune
if len(DF_tree.nodes[root]["GM"]) == k and DF_tree.nodes[root]["GBC"] > max_GBC:
return DF_tree.nodes[root]["GBC"], DF_tree, DF_tree.nodes[root]["GM"]
# stopping condition - if the size of group members equal to k or there are less than
# k - |GM| in the candidate list or the heuristic function plus the GBC is bellow the
# maximal GBC found then prune
if (
len(DF_tree.nodes[root]["GM"]) == k
or len(DF_tree.nodes[root]["CL"]) <= k - len(DF_tree.nodes[root]["GM"])
or DF_tree.nodes[root]["GBC"] + DF_tree.nodes[root]["heu"] <= max_GBC
):
return max_GBC, DF_tree, max_group

# finding the heuristic of both children
node_p, node_m, DF_tree = _heuristic(k, root, DF_tree, D, nodes, greedy)

# finding the child with the bigger heuristic + GBC and expand
# that node first if greedy then only expand the plus node
if greedy:
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
)

elif (
DF_tree.nodes[node_p]["GBC"] + DF_tree.nodes[node_p]["heu"]
> DF_tree.nodes[node_m]["GBC"] + DF_tree.nodes[node_m]["heu"]
):
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
)
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy
)
else:
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy
)
max_GBC, DF_tree, max_group = _dfbnb(
G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
)
return max_GBC, DF_tree, max_group

def _heuristic(k, root, DF_tree, D, nodes, greedy):
import numpy as np

# This helper function add two nodes to DF_tree - one left son and the
# other right son, finds their heuristic, CL, GBC, and GM
node_p = DF_tree.number_of_nodes() + 1
node_m = DF_tree.number_of_nodes() + 2
added_node = DF_tree.nodes[root]["CL"][0]

# adding the plus nude
DF_tree.add_nodes_from([(node_p, deepcopy(DF_tree.nodes[root]))])
DF_tree.nodes[node_p]["GM"].append(added_node)
DF_tree.nodes[node_p]["GBC"] += DF_tree.nodes[node_p]["cont"][added_node]
root_node = DF_tree.nodes[root]
for x in nodes:
for y in nodes:
dxvy = 0
dxyv = 0
dvxy = 0
if not (
root_node["sigma"][x][y] == 0
or root_node["sigma"][x][added_node] == 0
or root_node["sigma"][added_node][y] == 0
):
if D[x][added_node] == D[x][y] + D[y][added_node]:
dxyv = (
root_node["sigma"][x][y]
* root_node["sigma"][y][added_node]
/ root_node["sigma"][x][added_node]
)
if D[x][y] == D[x][added_node] + D[added_node][y]:
dxvy = (
root_node["sigma"][x][added_node]
* root_node["sigma"][added_node][y]
/ root_node["sigma"][x][y]
)
if D[added_node][y] == D[added_node][x] + D[x][y]:
dvxy = (
root_node["sigma"][added_node][x]
* root_node["sigma"][x][y]
/ root_node["sigma"][added_node][y]
)
DF_tree.nodes[node_p]["sigma"][x][y] = root_node["sigma"][x][y] * (1 - dxvy)
DF_tree.nodes[node_p]["betweenness"][x][y] = (
root_node["betweenness"][x][y] - root_node["betweenness"][x][y] * dxvy
)
if y != added_node:
DF_tree.nodes[node_p]["betweenness"][x][y] -= (
root_node["betweenness"][x][added_node] * dxyv
)
if x != added_node:
DF_tree.nodes[node_p]["betweenness"][x][y] -= (
root_node["betweenness"][added_node][y] * dvxy
)
CL = [
node
for _, node in sorted(
zip(np.diag(DF_tree.nodes[node_p]["betweenness"]), nodes), reverse=True
)
]
[CL.remove(m) for m in CL if m in DF_tree.nodes[node_p]["GM"]]
DF_tree.nodes[node_p]["CL"] = CL
DF_tree.nodes[node_p]["cont"] = dict(
zip(nodes, np.diag(DF_tree.nodes[node_p]["betweenness"]))
)
DF_tree.nodes[node_p]["heu"] = 0
for i in range(k - len(DF_tree.nodes[node_p]["GM"])):
DF_tree.nodes[node_p]["heu"] += DF_tree.nodes[node_p]["cont"][
DF_tree.nodes[node_p]["CL"][i]
]

# adding the minus node - don't insert the first node in the CL to GM
# Insert minus node only if isn't greedy type algorithm
if not greedy:
DF_tree.add_nodes_from([(node_m, deepcopy(DF_tree.nodes[root]))])
DF_tree.nodes[node_m]["CL"].pop(0)
DF_tree.nodes[node_m]["cont"].pop(added_node)
DF_tree.nodes[node_m]["heu"] = 0
for i in range(k - len(DF_tree.nodes[node_m]["GM"])):
DF_tree.nodes[node_m]["heu"] += DF_tree.nodes[node_m]["cont"][
DF_tree.nodes[node_m]["CL"][i]
]
else:
node_m = None

return node_p, node_m, DF_tree

[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.
The weight of an edge is treated as the length or distance between the two sides.

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.
https://doi.org/10.1145/2567948.2579356
"""
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)