"""Functions for finding and manipulating cliques.
Finding the largest clique in a graph is NP-complete problem, so most of
these algorithms have an exponential running time; for more information,
see the Wikipedia article on the clique problem [1]_.
.. [1] clique problem:: https://en.wikipedia.org/wiki/Clique_problem
"""
from collections import deque, defaultdict
from itertools import chain
from itertools import combinations
from itertools import islice
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"find_cliques",
"find_cliques_recursive",
"make_max_clique_graph",
"make_clique_bipartite",
"graph_clique_number",
"graph_number_of_cliques",
"node_clique_number",
"number_of_cliques",
"cliques_containing_node",
"enumerate_all_cliques",
"max_weight_clique",
]
[docs]@not_implemented_for("directed")
def enumerate_all_cliques(G):
"""Returns all cliques in an undirected graph.
This function returns an iterator over cliques, each of which is a
list of nodes. The iteration is ordered by cardinality of the
cliques: first all cliques of size one, then all cliques of size
two, etc.
Parameters
----------
G : NetworkX graph
An undirected graph.
Returns
-------
iterator
An iterator over cliques, each of which is a list of nodes in
`G`. The cliques are ordered according to size.
Notes
-----
To obtain a list of all cliques, use
`list(enumerate_all_cliques(G))`. However, be aware that in the
worst-case, the length of this list can be exponential in the number
of nodes in the graph (for example, when the graph is the complete
graph). This function avoids storing all cliques in memory by only
keeping current candidate node lists in memory during its search.
The implementation is adapted from the algorithm by Zhang, et
al. (2005) [1]_ to output all cliques discovered.
This algorithm ignores self-loops and parallel edges, since cliques
are not conventionally defined with such edges.
References
----------
.. [1] Yun Zhang, Abu-Khzam, F.N., Baldwin, N.E., Chesler, E.J.,
Langston, M.A., Samatova, N.F.,
"Genome-Scale Computational Approaches to Memory-Intensive
Applications in Systems Biology".
*Supercomputing*, 2005. Proceedings of the ACM/IEEE SC 2005
Conference, pp. 12, 12--18 Nov. 2005.
<https://doi.org/10.1109/SC.2005.29>.
"""
index = {}
nbrs = {}
for u in G:
index[u] = len(index)
# Neighbors of u that appear after u in the iteration order of G.
nbrs[u] = {v for v in G[u] if v not in index}
queue = deque(([u], sorted(nbrs[u], key=index.__getitem__)) for u in G)
# Loop invariants:
# 1. len(base) is nondecreasing.
# 2. (base + cnbrs) is sorted with respect to the iteration order of G.
# 3. cnbrs is a set of common neighbors of nodes in base.
while queue:
base, cnbrs = map(list, queue.popleft())
yield base
for i, u in enumerate(cnbrs):
# Use generators to reduce memory consumption.
queue.append(
(
chain(base, [u]),
filter(nbrs[u].__contains__, islice(cnbrs, i + 1, None)),
)
)
[docs]@not_implemented_for("directed")
def find_cliques(G, nodes=None):
"""Returns all maximal cliques in an undirected graph.
For each node *n*, a *maximal clique for n* is a largest complete
subgraph containing *n*. The largest maximal clique is sometimes
called the *maximum clique*.
This function returns an iterator over cliques, each of which is a
list of nodes. It is an iterative implementation, so should not
suffer from recursion depth issues.
This function accepts a list of `nodes` and only the maximal cliques
containing all of these `nodes` are returned. It can considerably speed up
the running time if some specific cliques are desired.
Parameters
----------
G : NetworkX graph
An undirected graph.
nodes : list, optional (default=None)
If provided, only yield *maximal cliques* containing all nodes in `nodes`.
If `nodes` isn't a clique itself, a ValueError is raised.
Returns
-------
iterator
An iterator over maximal cliques, each of which is a list of
nodes in `G`. If `nodes` is provided, only the maximal cliques
containing all the nodes in `nodes` are returned. The order of
cliques is arbitrary.
Raises
------
ValueError
If `nodes` is not a clique.
See Also
--------
find_cliques_recursive
A recursive version of the same algorithm.
Notes
-----
To obtain a list of all maximal cliques, use
`list(find_cliques(G))`. However, be aware that in the worst-case,
the length of this list can be exponential in the number of nodes in
the graph. This function avoids storing all cliques in memory by
only keeping current candidate node lists in memory during its search.
This implementation is based on the algorithm published by Bron and
Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi
(2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. It
essentially unrolls the recursion used in the references to avoid
issues of recursion stack depth (for a recursive implementation, see
:func:`find_cliques_recursive`).
This algorithm ignores self-loops and parallel edges, since cliques
are not conventionally defined with such edges.
References
----------
.. [1] Bron, C. and Kerbosch, J.
"Algorithm 457: finding all cliques of an undirected graph".
*Communications of the ACM* 16, 9 (Sep. 1973), 575--577.
<http://portal.acm.org/citation.cfm?doid=362342.362367>
.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
"The worst-case time complexity for generating all maximal
cliques and computational experiments",
*Theoretical Computer Science*, Volume 363, Issue 1,
Computing and Combinatorics,
10th Annual International Conference on
Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42
<https://doi.org/10.1016/j.tcs.2006.06.015>
.. [3] F. Cazals, C. Karande,
"A note on the problem of reporting maximal cliques",
*Theoretical Computer Science*,
Volume 407, Issues 1--3, 6 November 2008, Pages 564--568,
<https://doi.org/10.1016/j.tcs.2008.05.010>
"""
if len(G) == 0:
return
adj = {u: {v for v in G[u] if v != u} for u in G}
# Initialize Q with the given nodes and subg, cand with their nbrs
Q = nodes[:] if nodes is not None else []
cand = set(G)
for node in Q:
if node not in cand:
raise ValueError(f"The given `nodes` {nodes} do not form a clique")
cand &= adj[node]
if not cand:
yield Q[:]
return
subg = cand.copy()
stack = []
Q.append(None)
u = max(subg, key=lambda u: len(cand & adj[u]))
ext_u = cand - adj[u]
try:
while True:
if ext_u:
q = ext_u.pop()
cand.remove(q)
Q[-1] = q
adj_q = adj[q]
subg_q = subg & adj_q
if not subg_q:
yield Q[:]
else:
cand_q = cand & adj_q
if cand_q:
stack.append((subg, cand, ext_u))
Q.append(None)
subg = subg_q
cand = cand_q
u = max(subg, key=lambda u: len(cand & adj[u]))
ext_u = cand - adj[u]
else:
Q.pop()
subg, cand, ext_u = stack.pop()
except IndexError:
pass
# TODO Should this also be not implemented for directed graphs?
[docs]def find_cliques_recursive(G, nodes=None):
"""Returns all maximal cliques in a graph.
For each node *v*, a *maximal clique for v* is a largest complete
subgraph containing *v*. The largest maximal clique is sometimes
called the *maximum clique*.
This function returns an iterator over cliques, each of which is a
list of nodes. It is a recursive implementation, so may suffer from
recursion depth issues, but is included for pedagogical reasons.
For a non-recursive implementation, see :func:`find_cliques`.
This function accepts a list of `nodes` and only the maximal cliques
containing all of these `nodes` are returned. It can considerably speed up
the running time if some specific cliques are desired.
Parameters
----------
G : NetworkX graph
nodes : list, optional (default=None)
If provided, only yield *maximal cliques* containing all nodes in `nodes`.
If `nodes` isn't a clique itself, a ValueError is raised.
Returns
-------
iterator
An iterator over maximal cliques, each of which is a list of
nodes in `G`. If `nodes` is provided, only the maximal cliques
containing all the nodes in `nodes` are yielded. The order of
cliques is arbitrary.
Raises
------
ValueError
If `nodes` is not a clique.
See Also
--------
find_cliques
An iterative version of the same algorithm.
Notes
-----
To obtain a list of all maximal cliques, use
`list(find_cliques_recursive(G))`. However, be aware that in the
worst-case, the length of this list can be exponential in the number
of nodes in the graph. This function avoids storing all cliques in memory
by only keeping current candidate node lists in memory during its search.
This implementation is based on the algorithm published by Bron and
Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi
(2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. For a
non-recursive implementation, see :func:`find_cliques`.
This algorithm ignores self-loops and parallel edges, since cliques
are not conventionally defined with such edges.
References
----------
.. [1] Bron, C. and Kerbosch, J.
"Algorithm 457: finding all cliques of an undirected graph".
*Communications of the ACM* 16, 9 (Sep. 1973), 575--577.
<http://portal.acm.org/citation.cfm?doid=362342.362367>
.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
"The worst-case time complexity for generating all maximal
cliques and computational experiments",
*Theoretical Computer Science*, Volume 363, Issue 1,
Computing and Combinatorics,
10th Annual International Conference on
Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42
<https://doi.org/10.1016/j.tcs.2006.06.015>
.. [3] F. Cazals, C. Karande,
"A note on the problem of reporting maximal cliques",
*Theoretical Computer Science*,
Volume 407, Issues 1--3, 6 November 2008, Pages 564--568,
<https://doi.org/10.1016/j.tcs.2008.05.010>
"""
if len(G) == 0:
return iter([])
adj = {u: {v for v in G[u] if v != u} for u in G}
# Initialize Q with the given nodes and subg, cand with their nbrs
Q = nodes[:] if nodes is not None else []
cand_init = set(G)
for node in Q:
if node not in cand_init:
raise ValueError(f"The given `nodes` {nodes} do not form a clique")
cand_init &= adj[node]
if not cand_init:
return iter([Q])
subg_init = cand_init.copy()
def expand(subg, cand):
u = max(subg, key=lambda u: len(cand & adj[u]))
for q in cand - adj[u]:
cand.remove(q)
Q.append(q)
adj_q = adj[q]
subg_q = subg & adj_q
if not subg_q:
yield Q[:]
else:
cand_q = cand & adj_q
if cand_q:
yield from expand(subg_q, cand_q)
Q.pop()
return expand(subg_init, cand_init)
[docs]def make_max_clique_graph(G, create_using=None):
"""Returns the maximal clique graph of the given graph.
The nodes of the maximal clique graph of `G` are the cliques of
`G` and an edge joins two cliques if the cliques are not disjoint.
Parameters
----------
G : NetworkX graph
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
A graph whose nodes are the cliques of `G` and whose edges
join two cliques if they are not disjoint.
Notes
-----
This function behaves like the following code::
import networkx as nx
G = nx.make_clique_bipartite(G)
cliques = [v for v in G.nodes() if G.nodes[v]['bipartite'] == 0]
G = nx.bipartite.projected_graph(G, cliques)
G = nx.relabel_nodes(G, {-v: v - 1 for v in G})
It should be faster, though, since it skips all the intermediate
steps.
"""
if create_using is None:
B = G.__class__()
else:
B = nx.empty_graph(0, create_using)
cliques = list(enumerate(set(c) for c in find_cliques(G)))
# Add a numbered node for each clique.
B.add_nodes_from(i for i, c in cliques)
# Join cliques by an edge if they share a node.
clique_pairs = combinations(cliques, 2)
B.add_edges_from((i, j) for (i, c1), (j, c2) in clique_pairs if c1 & c2)
return B
[docs]def make_clique_bipartite(G, fpos=None, create_using=None, name=None):
"""Returns the bipartite clique graph corresponding to `G`.
In the returned bipartite graph, the "bottom" nodes are the nodes of
`G` and the "top" nodes represent the maximal cliques of `G`.
There is an edge from node *v* to clique *C* in the returned graph
if and only if *v* is an element of *C*.
Parameters
----------
G : NetworkX graph
An undirected graph.
fpos : bool
If True or not None, the returned graph will have an
additional attribute, `pos`, a dictionary mapping node to
position in the Euclidean plane.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
A bipartite graph whose "bottom" set is the nodes of the graph
`G`, whose "top" set is the cliques of `G`, and whose edges
join nodes of `G` to the cliques that contain them.
The nodes of the graph `G` have the node attribute
'bipartite' set to 1 and the nodes representing cliques
have the node attribute 'bipartite' set to 0, as is the
convention for bipartite graphs in NetworkX.
"""
B = nx.empty_graph(0, create_using)
B.clear()
# The "bottom" nodes in the bipartite graph are the nodes of the
# original graph, G.
B.add_nodes_from(G, bipartite=1)
for i, cl in enumerate(find_cliques(G)):
# The "top" nodes in the bipartite graph are the cliques. These
# nodes get negative numbers as labels.
name = -i - 1
B.add_node(name, bipartite=0)
B.add_edges_from((v, name) for v in cl)
return B
[docs]def graph_clique_number(G, cliques=None):
"""Returns the clique number of the graph.
The *clique number* of a graph is the size of the largest clique in
the graph.
Parameters
----------
G : NetworkX graph
An undirected graph.
cliques : list
A list of cliques, each of which is itself a list of nodes. If
not specified, the list of all cliques will be computed, as by
:func:`find_cliques`.
Returns
-------
int
The size of the largest clique in `G`.
Notes
-----
You should provide `cliques` if you have already computed the list
of maximal cliques, in order to avoid an exponential time search for
maximal cliques.
"""
if len(G.nodes) < 1:
return 0
if cliques is None:
cliques = find_cliques(G)
return max([len(c) for c in cliques] or [1])
[docs]def graph_number_of_cliques(G, cliques=None):
"""Returns the number of maximal cliques in the graph.
Parameters
----------
G : NetworkX graph
An undirected graph.
cliques : list
A list of cliques, each of which is itself a list of nodes. If
not specified, the list of all cliques will be computed, as by
:func:`find_cliques`.
Returns
-------
int
The number of maximal cliques in `G`.
Notes
-----
You should provide `cliques` if you have already computed the list
of maximal cliques, in order to avoid an exponential time search for
maximal cliques.
"""
if cliques is None:
cliques = list(find_cliques(G))
return len(cliques)
[docs]def node_clique_number(G, nodes=None, cliques=None, separate_nodes=False):
"""Returns the size of the largest maximal clique containing each given node.
Returns a single or list depending on input nodes.
An optional list of cliques can be input if already computed.
Parameters
----------
G : NetworkX graph
An undirected graph.
cliques : list, optional (default=None)
A list of cliques, each of which is itself a list of nodes.
If not specified, the list of all cliques will be computed
using :func:`find_cliques`.
Returns
-------
int or dict
If `nodes` is a single node, returns the size of the
largest maximal clique in `G` containing that node.
Otherwise return a dict keyed by node to the size
of the largest maximal clique containing that node.
See Also
--------
find_cliques
find_cliques yields the maximal cliques of G.
It accepts a `nodes` argument which restricts consideration to
maximal cliques containing all the given `nodes`.
The search for the cliques is optimized for `nodes`.
"""
if cliques is None:
if nodes is not None:
# Use ego_graph to decrease size of graph
# check for single node
if nodes in G:
return max(len(c) for c in find_cliques(nx.ego_graph(G, nodes)))
# handle multiple nodes
return {
n: max(len(c) for c in find_cliques(nx.ego_graph(G, n))) for n in nodes
}
# nodes is None--find all cliques
cliques = list(find_cliques(G))
# single node requested
if nodes in G:
return max(len(c) for c in cliques if nodes in c)
# multiple nodes requested
# preprocess all nodes (faster than one at a time for even 2 nodes)
size_for_n = defaultdict(int)
for c in cliques:
size_of_c = len(c)
for n in c:
if size_for_n[n] < size_of_c:
size_for_n[n] = size_of_c
if nodes is None:
return size_for_n
return {n: size_for_n[n] for n in nodes}
[docs]def number_of_cliques(G, nodes=None, cliques=None):
"""Returns the number of maximal cliques for each node.
Returns a single or list depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques = list(find_cliques(G))
if nodes is None:
nodes = list(G.nodes()) # none, get entire graph
if not isinstance(nodes, list): # check for a list
v = nodes
# assume it is a single value
numcliq = len([1 for c in cliques if v in c])
else:
numcliq = {}
for v in nodes:
numcliq[v] = len([1 for c in cliques if v in c])
return numcliq
[docs]def cliques_containing_node(G, nodes=None, cliques=None):
"""Returns a list of cliques containing the given node.
Returns a single list or list of lists depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques = list(find_cliques(G))
if nodes is None:
nodes = list(G.nodes()) # none, get entire graph
if not isinstance(nodes, list): # check for a list
v = nodes
# assume it is a single value
vcliques = [c for c in cliques if v in c]
else:
vcliques = {}
for v in nodes:
vcliques[v] = [c for c in cliques if v in c]
return vcliques
class MaxWeightClique:
"""A class for the maximum weight clique algorithm.
This class is a helper for the `max_weight_clique` function. The class
should not normally be used directly.
Parameters
----------
G : NetworkX graph
The undirected graph for which a maximum weight clique is sought
weight : string or None, optional (default='weight')
The node attribute that holds the integer value used as a weight.
If None, then each node has weight 1.
Attributes
----------
G : NetworkX graph
The undirected graph for which a maximum weight clique is sought
node_weights: dict
The weight of each node
incumbent_nodes : list
The nodes of the incumbent clique (the best clique found so far)
incumbent_weight: int
The weight of the incumbent clique
"""
def __init__(self, G, weight):
self.G = G
self.incumbent_nodes = []
self.incumbent_weight = 0
if weight is None:
self.node_weights = {v: 1 for v in G.nodes()}
else:
for v in G.nodes():
if weight not in G.nodes[v]:
errmsg = f"Node {v!r} does not have the requested weight field."
raise KeyError(errmsg)
if not isinstance(G.nodes[v][weight], int):
errmsg = f"The {weight!r} field of node {v!r} is not an integer."
raise ValueError(errmsg)
self.node_weights = {v: G.nodes[v][weight] for v in G.nodes()}
def update_incumbent_if_improved(self, C, C_weight):
"""Update the incumbent if the node set C has greater weight.
C is assumed to be a clique.
"""
if C_weight > self.incumbent_weight:
self.incumbent_nodes = C[:]
self.incumbent_weight = C_weight
def greedily_find_independent_set(self, P):
"""Greedily find an independent set of nodes from a set of
nodes P."""
independent_set = []
P = P[:]
while P:
v = P[0]
independent_set.append(v)
P = [w for w in P if v != w and not self.G.has_edge(v, w)]
return independent_set
def find_branching_nodes(self, P, target):
"""Find a set of nodes to branch on."""
residual_wt = {v: self.node_weights[v] for v in P}
total_wt = 0
P = P[:]
while P:
independent_set = self.greedily_find_independent_set(P)
min_wt_in_class = min(residual_wt[v] for v in independent_set)
total_wt += min_wt_in_class
if total_wt > target:
break
for v in independent_set:
residual_wt[v] -= min_wt_in_class
P = [v for v in P if residual_wt[v] != 0]
return P
def expand(self, C, C_weight, P):
"""Look for the best clique that contains all the nodes in C and zero or
more of the nodes in P, backtracking if it can be shown that no such
clique has greater weight than the incumbent.
"""
self.update_incumbent_if_improved(C, C_weight)
branching_nodes = self.find_branching_nodes(P, self.incumbent_weight - C_weight)
while branching_nodes:
v = branching_nodes.pop()
P.remove(v)
new_C = C + [v]
new_C_weight = C_weight + self.node_weights[v]
new_P = [w for w in P if self.G.has_edge(v, w)]
self.expand(new_C, new_C_weight, new_P)
def find_max_weight_clique(self):
"""Find a maximum weight clique."""
# Sort nodes in reverse order of degree for speed
nodes = sorted(self.G.nodes(), key=lambda v: self.G.degree(v), reverse=True)
nodes = [v for v in nodes if self.node_weights[v] > 0]
self.expand([], 0, nodes)
[docs]@not_implemented_for("directed")
def max_weight_clique(G, weight="weight"):
"""Find a maximum weight clique in G.
A *clique* in a graph is a set of nodes such that every two distinct nodes
are adjacent. The *weight* of a clique is the sum of the weights of its
nodes. A *maximum weight clique* of graph G is a clique C in G such that
no clique in G has weight greater than the weight of C.
Parameters
----------
G : NetworkX graph
Undirected graph
weight : string or None, optional (default='weight')
The node attribute that holds the integer value used as a weight.
If None, then each node has weight 1.
Returns
-------
clique : list
the nodes of a maximum weight clique
weight : int
the weight of a maximum weight clique
Notes
-----
The implementation is recursive, and therefore it may run into recursion
depth issues if G contains a clique whose number of nodes is close to the
recursion depth limit.
At each search node, the algorithm greedily constructs a weighted
independent set cover of part of the graph in order to find a small set of
nodes on which to branch. The algorithm is very similar to the algorithm
of Tavares et al. [1]_, other than the fact that the NetworkX version does
not use bitsets. This style of algorithm for maximum weight clique (and
maximum weight independent set, which is the same problem but on the
complement graph) has a decades-long history. See Algorithm B of Warren
and Hicks [2]_ and the references in that paper.
References
----------
.. [1] Tavares, W.A., Neto, M.B.C., Rodrigues, C.D., Michelon, P.: Um
algoritmo de branch and bound para o problema da clique máxima
ponderada. Proceedings of XLVII SBPO 1 (2015).
.. [2] Warrent, Jeffrey S, Hicks, Illya V.: Combinatorial Branch-and-Bound
for the Maximum Weight Independent Set Problem. Technical Report,
Texas A&M University (2016).
"""
mwc = MaxWeightClique(G, weight)
mwc.find_max_weight_clique()
return mwc.incumbent_nodes, mwc.incumbent_weight