Source code for networkx.generators.community

"""Generators for classes of graphs used in studying social networks."""
import itertools
import math

import networkx as nx
from networkx.utils import py_random_state

__all__ = [
"caveman_graph",
"connected_caveman_graph",
"relaxed_caveman_graph",
"random_partition_graph",
"planted_partition_graph",
"gaussian_random_partition_graph",
"ring_of_cliques",
"windmill_graph",
"stochastic_block_model",
"LFR_benchmark_graph",
]

[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def caveman_graph(l, k):
"""Returns a caveman graph of l cliques of size k.

Parameters
----------
l : int
Number of cliques
k : int
Size of cliques

Returns
-------
G : NetworkX Graph
caveman graph

Notes
-----
This returns an undirected graph, it can be converted to a directed
graph using :func:nx.to_directed, or a multigraph using
nx.MultiGraph(nx.caveman_graph(l, k)). Only the undirected version is
described in [1]_ and it is unclear which of the directed
generalizations is most useful.

Examples
--------
>>> G = nx.caveman_graph(3, 3)

--------

connected_caveman_graph

References
----------
.. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
Amer. J. Soc. 105, 493-527, 1999.
"""
# l disjoint cliques of size k
G = nx.empty_graph(l * k)
if k > 1:
for start in range(0, l * k, k):
edges = itertools.combinations(range(start, start + k), 2)
return G

[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def connected_caveman_graph(l, k):
"""Returns a connected caveman graph of l cliques of size k.

The connected caveman graph is formed by creating n cliques of size
k, then a single edge in each clique is rewired to a node in an

Parameters
----------
l : int
number of cliques
k : int
size of cliques (k at least 2 or NetworkXError is raised)

Returns
-------
G : NetworkX Graph
connected caveman graph

Raises
------
NetworkXError
If the size of cliques k is smaller than 2.

Notes
-----
This returns an undirected graph, it can be converted to a directed
graph using :func:nx.to_directed, or a multigraph using
nx.MultiGraph(nx.caveman_graph(l, k)). Only the undirected version is
described in [1]_ and it is unclear which of the directed
generalizations is most useful.

Examples
--------
>>> G = nx.connected_caveman_graph(3, 3)

References
----------
.. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
Amer. J. Soc. 105, 493-527, 1999.
"""
if k < 2:
raise nx.NetworkXError(
"The size of cliques in a connected caveman graph must be at least 2."
)

G = nx.caveman_graph(l, k)
for start in range(0, l * k, k):
G.remove_edge(start, start + 1)
G.add_edge(start, (start - 1) % (l * k))
return G

[docs]
@py_random_state(3)
@nx._dispatchable(graphs=None, returns_graph=True)
def relaxed_caveman_graph(l, k, p, seed=None):
"""Returns a relaxed caveman graph.

A relaxed caveman graph starts with l cliques of size k.  Edges are
then randomly rewired with probability p to link different cliques.

Parameters
----------
l : int
Number of groups
k : int
Size of cliques
p : float
Probability of rewiring each edge.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : NetworkX Graph
Relaxed Caveman Graph

Raises
------
NetworkXError
If p is not in [0,1]

Examples
--------
>>> G = nx.relaxed_caveman_graph(2, 3, 0.1, seed=42)

References
----------
.. [1] Santo Fortunato, Community Detection in Graphs,
Physics Reports Volume 486, Issues 3-5, February 2010, Pages 75-174.
https://arxiv.org/abs/0906.0612
"""
G = nx.caveman_graph(l, k)
nodes = list(G)
for u, v in G.edges():
if seed.random() < p:  # rewire the edge
x = seed.choice(nodes)
if G.has_edge(u, x):
continue
G.remove_edge(u, v)
return G

[docs]
@py_random_state(3)
@nx._dispatchable(graphs=None, returns_graph=True)
def random_partition_graph(sizes, p_in, p_out, seed=None, directed=False):
"""Returns the random partition graph with a partition of sizes.

A partition graph is a graph of communities with sizes defined by
s in sizes. Nodes in the same group are connected with probability
p_in and nodes of different groups are connected with probability
p_out.

Parameters
----------
sizes : list of ints
Sizes of groups
p_in : float
probability of edges with in groups
p_out : float
probability of edges between groups
directed : boolean optional, default=False
Whether to create a directed graph
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : NetworkX Graph or DiGraph
random partition graph of size sum(gs)

Raises
------
NetworkXError
If p_in or p_out is not in [0,1]

Examples
--------
>>> G = nx.random_partition_graph([10, 10, 10], 0.25, 0.01)
>>> len(G)
30
>>> partition = G.graph["partition"]
>>> len(partition)
3

Notes
-----
This is a generalization of the planted-l-partition described in
[1]_.  It allows for the creation of groups of any size.

The partition is store as a graph attribute 'partition'.

References
----------
.. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612
"""
# Use geometric method for O(n+m) complexity algorithm
# partition = nx.community_sets(nx.get_node_attributes(G, 'affiliation'))
if not 0.0 <= p_in <= 1.0:
raise nx.NetworkXError("p_in must be in [0,1]")
if not 0.0 <= p_out <= 1.0:
raise nx.NetworkXError("p_out must be in [0,1]")

# create connection matrix
num_blocks = len(sizes)
p = [[p_out for s in range(num_blocks)] for r in range(num_blocks)]
for r in range(num_blocks):
p[r][r] = p_in

return stochastic_block_model(
sizes,
p,
nodelist=None,
seed=seed,
directed=directed,
selfloops=False,
sparse=True,
)

[docs]
@py_random_state(4)
@nx._dispatchable(graphs=None, returns_graph=True)
def planted_partition_graph(l, k, p_in, p_out, seed=None, directed=False):
"""Returns the planted l-partition graph.

This model partitions a graph with n=l*k vertices in
l groups with k vertices each. Vertices of the same
group are linked with a probability p_in, and vertices
of different groups are linked with probability p_out.

Parameters
----------
l : int
Number of groups
k : int
Number of vertices in each group
p_in : float
probability of connecting vertices within a group
p_out : float
probability of connected vertices between groups
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.
directed : bool,optional (default=False)
If True return a directed graph

Returns
-------
G : NetworkX Graph or DiGraph
planted l-partition graph

Raises
------
NetworkXError
If p_in, p_out are not in [0, 1]

Examples
--------
>>> G = nx.planted_partition_graph(4, 3, 0.5, 0.1, seed=42)

--------
random_partition_model

References
----------
.. [1] A. Condon, R.M. Karp, Algorithms for graph partitioning
on the planted partition model,
Random Struct. Algor. 18 (2001) 116-140.

.. [2] Santo Fortunato 'Community Detection in Graphs' Physical Reports
Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612
"""
return random_partition_graph([k] * l, p_in, p_out, seed=seed, directed=directed)

[docs]
@py_random_state(6)
@nx._dispatchable(graphs=None, returns_graph=True)
def gaussian_random_partition_graph(n, s, v, p_in, p_out, directed=False, seed=None):
"""Generate a Gaussian random partition graph.

A Gaussian random partition graph is created by creating k partitions
each with a size drawn from a normal distribution with mean s and variance
s/v. Nodes are connected within clusters with probability p_in and
between clusters with probability p_out[1]

Parameters
----------
n : int
Number of nodes in the graph
s : float
Mean cluster size
v : float
Shape parameter. The variance of cluster size distribution is s/v.
p_in : float
Probability of intra cluster connection.
p_out : float
Probability of inter cluster connection.
directed : boolean, optional default=False
Whether to create a directed graph or not
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : NetworkX Graph or DiGraph
gaussian random partition graph

Raises
------
NetworkXError
If s is > n
If p_in or p_out is not in [0,1]

Notes
-----
Note the number of partitions is dependent on s,v and n, and that the
last partition may be considerably smaller, as it is sized to simply
fill out the nodes [1]

--------
random_partition_graph

Examples
--------
>>> G = nx.gaussian_random_partition_graph(100, 10, 10, 0.25, 0.1)
>>> len(G)
100

References
----------
.. [1] Ulrik Brandes, Marco Gaertler, Dorothea Wagner,
Experiments on Graph Clustering Algorithms,
In the proceedings of the 11th Europ. Symp. Algorithms, 2003.
"""
if s > n:
raise nx.NetworkXError("s must be <= n")
assigned = 0
sizes = []
while True:
size = int(seed.gauss(s, s / v + 0.5))
if size < 1:  # how to handle 0 or negative sizes?
continue
if assigned + size >= n:
sizes.append(n - assigned)
break
assigned += size
sizes.append(size)
return random_partition_graph(sizes, p_in, p_out, seed=seed, directed=directed)

[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def ring_of_cliques(num_cliques, clique_size):
"""Defines a "ring of cliques" graph.

A ring of cliques graph is consisting of cliques, connected through single
links. Each clique is a complete graph.

Parameters
----------
num_cliques : int
Number of cliques
clique_size : int
Size of cliques

Returns
-------
G : NetworkX Graph
ring of cliques graph

Raises
------
NetworkXError
If the number of cliques is lower than 2 or
if the size of cliques is smaller than 2.

Examples
--------
>>> G = nx.ring_of_cliques(8, 4)

--------
connected_caveman_graph

Notes
-----
The connected_caveman_graph graph removes a link from each clique to
connect it with the next clique. Instead, the ring_of_cliques graph
"""
if num_cliques < 2:
raise nx.NetworkXError("A ring of cliques must have at least two cliques")
if clique_size < 2:
raise nx.NetworkXError("The cliques must have at least two nodes")

G = nx.Graph()
for i in range(num_cliques):
edges = itertools.combinations(
range(i * clique_size, i * clique_size + clique_size), 2
)
i * clique_size + 1, (i + 1) * clique_size % (num_cliques * clique_size)
)
return G

[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def windmill_graph(n, k):
"""Generate a windmill graph.
A windmill graph is a graph of n cliques each of size k that are all
joined at one node.
It can be thought of as taking a disjoint union of n cliques of size k,
selecting one point from each, and contracting all of the selected points.
Alternatively, one could generate n cliques of size k-1 and one node
that is connected to all other nodes in the graph.

Parameters
----------
n : int
Number of cliques
k : int
Size of cliques

Returns
-------
G : NetworkX Graph
windmill graph with n cliques of size k

Raises
------
NetworkXError
If the number of cliques is less than two
If the size of the cliques are less than two

Examples
--------
>>> G = nx.windmill_graph(4, 5)

Notes
-----
The node labeled 0 will be the node connected to all other nodes.
Note that windmill graphs are usually denoted Wd(k,n), so the parameters
are in the opposite order as the parameters of this method.
"""
if n < 2:
msg = "A windmill graph must have at least two cliques"
raise nx.NetworkXError(msg)
if k < 2:
raise nx.NetworkXError("The cliques must have at least two nodes")

G = nx.disjoint_union_all(
itertools.chain(
[nx.complete_graph(k)], (nx.complete_graph(k - 1) for _ in range(n - 1))
)
)
G.add_edges_from((0, i) for i in range(k, G.number_of_nodes()))
return G

[docs]
@py_random_state(3)
@nx._dispatchable(graphs=None, returns_graph=True)
def stochastic_block_model(
sizes, p, nodelist=None, seed=None, directed=False, selfloops=False, sparse=True
):
"""Returns a stochastic block model graph.

This model partitions the nodes in blocks of arbitrary sizes, and places
edges between pairs of nodes independently, with a probability that depends
on the blocks.

Parameters
----------
sizes : list of ints
Sizes of blocks
p : list of list of floats
Element (r,s) gives the density of edges going from the nodes
of group r to nodes of group s.
p must match the number of groups (len(sizes) == len(p)),
and it must be symmetric if the graph is undirected.
nodelist : list, optional
The block tags are assigned according to the node identifiers
in nodelist. If nodelist is None, then the ordering is the
range [0,sum(sizes)-1].
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.
directed : boolean optional, default=False
Whether to create a directed graph or not.
selfloops : boolean optional, default=False
Whether to include self-loops or not.
sparse: boolean optional, default=True
Use the sparse heuristic to speed up the generator.

Returns
-------
g : NetworkX Graph or DiGraph
Stochastic block model graph of size sum(sizes)

Raises
------
NetworkXError
If probabilities are not in [0,1].
If the probability matrix is not square (directed case).
If the probability matrix is not symmetric (undirected case).
If the sizes list does not match nodelist or the probability matrix.
If nodelist contains duplicate.

Examples
--------
>>> sizes = [75, 75, 300]
>>> probs = [[0.25, 0.05, 0.02], [0.05, 0.35, 0.07], [0.02, 0.07, 0.40]]
>>> g = nx.stochastic_block_model(sizes, probs, seed=0)
>>> len(g)
450
>>> H = nx.quotient_graph(g, g.graph["partition"], relabel=True)
>>> for v in H.nodes(data=True):
...     print(round(v[1]["density"], 3))
0.245
0.348
0.405
>>> for v in H.edges(data=True):
...     print(round(1.0 * v[2]["weight"] / (sizes[v[0]] * sizes[v[1]]), 3))
0.051
0.022
0.07

--------
random_partition_graph
planted_partition_graph
gaussian_random_partition_graph
gnp_random_graph

References
----------
.. [1] Holland, P. W., Laskey, K. B., & Leinhardt, S.,
"Stochastic blockmodels: First steps",
Social networks, 5(2), 109-137, 1983.
"""
# Check if dimensions match
if len(sizes) != len(p):
raise nx.NetworkXException("'sizes' and 'p' do not match.")
# Check for probability symmetry (undirected) and shape (directed)
for row in p:
if len(p) != len(row):
raise nx.NetworkXException("'p' must be a square matrix.")
if not directed:
p_transpose = [list(i) for i in zip(*p)]
for i in zip(p, p_transpose):
for j in zip(i[0], i[1]):
if abs(j[0] - j[1]) > 1e-08:
raise nx.NetworkXException("'p' must be symmetric.")
# Check for probability range
for row in p:
for prob in row:
if prob < 0 or prob > 1:
raise nx.NetworkXException("Entries of 'p' not in [0,1].")
# Check for nodelist consistency
if nodelist is not None:
if len(nodelist) != sum(sizes):
raise nx.NetworkXException("'nodelist' and 'sizes' do not match.")
if len(nodelist) != len(set(nodelist)):
raise nx.NetworkXException("nodelist contains duplicate.")
else:
nodelist = range(sum(sizes))

# Setup the graph conditionally to the directed switch.
block_range = range(len(sizes))
if directed:
g = nx.DiGraph()
block_iter = itertools.product(block_range, block_range)
else:
g = nx.Graph()
block_iter = itertools.combinations_with_replacement(block_range, 2)
# Split nodelist in a partition (list of sets).
size_cumsum = [sum(sizes[0:x]) for x in range(len(sizes) + 1)]
g.graph["partition"] = [
set(nodelist[size_cumsum[x] : size_cumsum[x + 1]])
for x in range(len(size_cumsum) - 1)
]
# Setup nodes and graph name
for block_id, nodes in enumerate(g.graph["partition"]):
for node in nodes:

g.name = "stochastic_block_model"

# Test for edge existence
parts = g.graph["partition"]
for i, j in block_iter:
if i == j:
if directed:
if selfloops:
edges = itertools.product(parts[i], parts[i])
else:
edges = itertools.permutations(parts[i], 2)
else:
edges = itertools.combinations(parts[i], 2)
if selfloops:
edges = itertools.chain(edges, zip(parts[i], parts[i]))
for e in edges:
if seed.random() < p[i][j]:
else:
edges = itertools.product(parts[i], parts[j])
if sparse:
if p[i][j] == 1:  # Test edges cases p_ij = 0 or 1
for e in edges:
elif p[i][j] > 0:
while True:
try:
logrand = math.log(seed.random())
skip = math.floor(logrand / math.log(1 - p[i][j]))
# consume "skip" edges
next(itertools.islice(edges, skip, skip), None)
e = next(edges)
except StopIteration:
break
else:
for e in edges:
if seed.random() < p[i][j]:
return g

def _zipf_rv_below(gamma, xmin, threshold, seed):
"""Returns a random value chosen from the bounded Zipf distribution.

Repeatedly draws values from the Zipf distribution until the
threshold is met, then returns that value.
"""
result = nx.utils.zipf_rv(gamma, xmin, seed)
while result > threshold:
result = nx.utils.zipf_rv(gamma, xmin, seed)
return result

def _powerlaw_sequence(gamma, low, high, condition, length, max_iters, seed):
"""Returns a list of numbers obeying a constrained power law distribution.

gamma and low are the parameters for the Zipf distribution.

high is the maximum allowed value for values draw from the Zipf
distribution. For more information, see :func:_zipf_rv_below.

condition and length are Boolean-valued functions on
lists. While generating the list, random values are drawn and
appended to the list until length is satisfied by the created
list. Once condition is satisfied, the sequence generated in
this way is returned.

max_iters indicates the number of times to generate a list
satisfying length. If the number of iterations exceeds this
value, :exc:~networkx.exception.ExceededMaxIterations is raised.

seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.
"""
for i in range(max_iters):
seq = []
while not length(seq):
seq.append(_zipf_rv_below(gamma, low, high, seed))
if condition(seq):
return seq
raise nx.ExceededMaxIterations("Could not create power law sequence")

def _hurwitz_zeta(x, q, tolerance):
"""The Hurwitz zeta function, or the Riemann zeta function of two arguments.

x must be greater than one and q must be positive.

This function repeatedly computes subsequent partial sums until
convergence, as decided by tolerance.
"""
z = 0
z_prev = -float("inf")
k = 0
while abs(z - z_prev) > tolerance:
z_prev = z
z += 1 / ((k + q) ** x)
k += 1
return z

def _generate_min_degree(gamma, average_degree, max_degree, tolerance, max_iters):
"""Returns a minimum degree from the given average degree."""
# Defines zeta function whether or not Scipy is available
try:
from scipy.special import zeta
except ImportError:

def zeta(x, q):
return _hurwitz_zeta(x, q, tolerance)

min_deg_top = max_degree
min_deg_bot = 1
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
itrs = 0
mid_avg_deg = 0
while abs(mid_avg_deg - average_degree) > tolerance:
if itrs > max_iters:
raise nx.ExceededMaxIterations("Could not match average_degree")
mid_avg_deg = 0
for x in range(int(min_deg_mid), max_degree + 1):
mid_avg_deg += (x ** (-gamma + 1)) / zeta(gamma, min_deg_mid)
if mid_avg_deg > average_degree:
min_deg_top = min_deg_mid
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
else:
min_deg_bot = min_deg_mid
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
itrs += 1
# return int(min_deg_mid + 0.5)
return round(min_deg_mid)

def _generate_communities(degree_seq, community_sizes, mu, max_iters, seed):
"""Returns a list of sets, each of which represents a community.

degree_seq is the degree sequence that must be met by the
graph.

community_sizes is the community size distribution that must be
met by the generated list of sets.

mu is a float in the interval [0, 1] indicating the fraction of
intra-community edges incident to each node.

max_iters is the number of times to try to add a node to a
community. This must be greater than the length of
degree_seq, otherwise this function will always fail. If
the number of iterations exceeds this value,
:exc:~networkx.exception.ExceededMaxIterations is raised.

seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

The communities returned by this are sets of integers in the set {0,
..., *n* - 1}, where *n* is the length of degree_seq.

"""
# This assumes the nodes in the graph will be natural numbers.
result = [set() for _ in community_sizes]
n = len(degree_seq)
free = list(range(n))
for i in range(max_iters):
v = free.pop()
c = seed.choice(range(len(community_sizes)))
# s = int(degree_seq[v] * (1 - mu) + 0.5)
s = round(degree_seq[v] * (1 - mu))
# If the community is large enough, add the node to the chosen
# community. Otherwise, return it to the list of unaffiliated
# nodes.
if s < community_sizes[c]:
else:
free.append(v)
# If the community is too big, remove a node from it.
if len(result[c]) > community_sizes[c]:
free.append(result[c].pop())
if not free:
return result
msg = "Could not assign communities; try increasing min_community"
raise nx.ExceededMaxIterations(msg)

[docs]
@py_random_state(11)
@nx._dispatchable(graphs=None, returns_graph=True)
def LFR_benchmark_graph(
n,
tau1,
tau2,
mu,
average_degree=None,
min_degree=None,
max_degree=None,
min_community=None,
max_community=None,
tol=1.0e-7,
max_iters=500,
seed=None,
):
r"""Returns the LFR benchmark graph.

This algorithm proceeds as follows:

1) Find a degree sequence with a power law distribution, and minimum
value min_degree, which has approximate average degree
average_degree. This is accomplished by either

a) specifying min_degree and not average_degree,
b) specifying average_degree and not min_degree, in which
case a suitable minimum degree will be found.

max_degree can also be specified, otherwise it will be set to
n. Each node *u* will have $\mu \mathrm{deg}(u)$ edges
joining it to nodes in communities other than its own and $(1 - \mu) \mathrm{deg}(u)$ edges joining it to nodes in its own
community.
2) Generate community sizes according to a power law distribution
with exponent tau2. If min_community and
max_community are not specified they will be selected to be
min_degree and max_degree, respectively.  Community sizes
are generated until the sum of their sizes equals n.
3) Each node will be randomly assigned a community with the
condition that the community is large enough for the node's
intra-community degree, $(1 - \mu) \mathrm{deg}(u)$ as
described in step 2. If a community grows too large, a random node
will be selected for reassignment to a new community, until all
nodes have been assigned a community.
4) Each node *u* then adds $(1 - \mu) \mathrm{deg}(u)$
intra-community edges and $\mu \mathrm{deg}(u)$ inter-community
edges.

Parameters
----------
n : int
Number of nodes in the created graph.

tau1 : float
Power law exponent for the degree distribution of the created
graph. This value must be strictly greater than one.

tau2 : float
Power law exponent for the community size distribution in the
created graph. This value must be strictly greater than one.

mu : float
Fraction of inter-community edges incident to each node. This
value must be in the interval [0, 1].

average_degree : float
Desired average degree of nodes in the created graph. This value
must be in the interval [0, *n*]. Exactly one of this and
min_degree must be specified, otherwise a
:exc:NetworkXError is raised.

min_degree : int
Minimum degree of nodes in the created graph. This value must be
in the interval [0, *n*]. Exactly one of this and
average_degree must be specified, otherwise a
:exc:NetworkXError is raised.

max_degree : int
Maximum degree of nodes in the created graph. If not specified,
this is set to n, the total number of nodes in the graph.

min_community : int
Minimum size of communities in the graph. If not specified, this
is set to min_degree.

max_community : int
Maximum size of communities in the graph. If not specified, this
is set to n, the total number of nodes in the graph.

tol : float
Tolerance when comparing floats, specifically when comparing
average degree values.

max_iters : int
Maximum number of iterations to try to create the community sizes,
degree distribution, and community affiliations.

seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : NetworkX graph
The LFR benchmark graph generated according to the specified
parameters.

Each node in the graph has a node attribute 'community' that
stores the community (that is, the set of nodes) that includes
it.

Raises
------
NetworkXError
If any of the parameters do not meet their upper and lower bounds:

- tau1 and tau2 must be strictly greater than 1.
- mu must be in [0, 1].
- max_degree must be in {1, ..., *n*}.
- min_community and max_community must be in {0, ...,
*n*}.

If not exactly one of average_degree and min_degree is
specified.

If min_degree is not specified and a suitable min_degree
cannot be found.

ExceededMaxIterations
If a valid degree sequence cannot be created within
max_iters number of iterations.

If a valid set of community sizes cannot be created within
max_iters number of iterations.

If a valid community assignment cannot be created within 10 *
n * max_iters number of iterations.

Examples
--------
Basic usage::

>>> from networkx.generators.community import LFR_benchmark_graph
>>> n = 250
>>> tau1 = 3
>>> tau2 = 1.5
>>> mu = 0.1
>>> G = LFR_benchmark_graph(
...     n, tau1, tau2, mu, average_degree=5, min_community=20, seed=10
... )

Continuing the example above, you can get the communities from the
node attributes of the graph::

>>> communities = {frozenset(G.nodes[v]["community"]) for v in G}

Notes
-----
This algorithm differs slightly from the original way it was
presented in [1].

1) Rather than connecting the graph via a configuration model then
rewiring to match the intra-community and inter-community
degrees, we do this wiring explicitly at the end, which should be
equivalent.
2) The code posted on the author's website [2] calculates the random
power law distributed variables and their average using
continuous approximations, whereas we use the discrete
distributions here as both degree and community size are
discrete.

Though the authors describe the algorithm as quite robust, testing
during development indicates that a somewhat narrower parameter set
is likely to successfully produce a graph. Some suggestions have
been provided in the event of exceptions.

References
----------
.. [1] "Benchmark graphs for testing community detection algorithms",
Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi,
Phys. Rev. E 78, 046110 2008
.. [2] https://www.santofortunato.net/resources

"""
# Perform some basic parameter validation.
if not tau1 > 1:
raise nx.NetworkXError("tau1 must be greater than one")
if not tau2 > 1:
raise nx.NetworkXError("tau2 must be greater than one")
if not 0 <= mu <= 1:
raise nx.NetworkXError("mu must be in the interval [0, 1]")

# Validate parameters for generating the degree sequence.
if max_degree is None:
max_degree = n
elif not 0 < max_degree <= n:
raise nx.NetworkXError("max_degree must be in the interval (0, n]")
if not ((min_degree is None) ^ (average_degree is None)):
raise nx.NetworkXError(
"Must assign exactly one of min_degree and average_degree"
)
if min_degree is None:
min_degree = _generate_min_degree(
tau1, average_degree, max_degree, tol, max_iters
)

# Generate a degree sequence with a power law distribution.
low, high = min_degree, max_degree

def condition(seq):
return sum(seq) % 2 == 0

def length(seq):
return len(seq) >= n

deg_seq = _powerlaw_sequence(tau1, low, high, condition, length, max_iters, seed)

# Validate parameters for generating the community size sequence.
if min_community is None:
min_community = min(deg_seq)
if max_community is None:
max_community = max(deg_seq)

# Generate a community size sequence with a power law distribution.
#
# TODO The original code incremented the number of iterations each
# time a new Zipf random value was drawn from the distribution. This
# differed from the way the number of iterations was incremented in
# _powerlaw_degree_sequence, so this code was changed to match
# that one. As a result, this code is allowed many more chances to
# generate a valid community size sequence.
low, high = min_community, max_community

def condition(seq):
return sum(seq) == n

def length(seq):
return sum(seq) >= n

comms = _powerlaw_sequence(tau2, low, high, condition, length, max_iters, seed)

# Generate the communities based on the given degree sequence and
# community sizes.
max_iters *= 10 * n
communities = _generate_communities(deg_seq, comms, mu, max_iters, seed)

# Finally, generate the benchmark graph based on the given
# communities, joining nodes according to the intra- and
# inter-community degrees.
G = nx.Graph()