# Source code for networkx.generators.degree_seq

"""Generate graphs with a given degree sequence or expected degree sequence.
"""

import heapq
import math
from itertools import chain, combinations, zip_longest
from operator import itemgetter

import networkx as nx
from networkx.utils import py_random_state, random_weighted_sample

__all__ = [
"configuration_model",
"directed_configuration_model",
"expected_degree_graph",
"havel_hakimi_graph",
"directed_havel_hakimi_graph",
"degree_sequence_tree",
"random_degree_sequence_graph",
]

chaini = chain.from_iterable

def _to_stublist(degree_sequence):
"""Returns a list of degree-repeated node numbers.

degree_sequence is a list of nonnegative integers representing
the degrees of nodes in a graph.

This function returns a list of node numbers with multiplicities
according to the given degree sequence. For example, if the first
element of degree_sequence is 3, then the first node number,
0, will appear at the head of the returned list three times. The
node numbers are assumed to be the numbers zero through
len(degree_sequence) - 1.

Examples
--------

>>> degree_sequence = [1, 2, 3]
>>> _to_stublist(degree_sequence)
[0, 1, 1, 2, 2, 2]

If a zero appears in the sequence, that means the node exists but
has degree zero, so that number will be skipped in the returned
list::

>>> degree_sequence = [2, 0, 1]
>>> _to_stublist(degree_sequence)
[0, 0, 2]

"""
return list(chaini([n] * d for n, d in enumerate(degree_sequence)))

def _configuration_model(
deg_sequence, create_using, directed=False, in_deg_sequence=None, seed=None
):
"""Helper function for generating either undirected or directed
configuration model graphs.

deg_sequence is a list of nonnegative integers representing the
degree of the node whose label is the index of the list element.

create_using see :func:~networkx.empty_graph.

directed and in_deg_sequence are required if you want the
returned graph to be generated using the directed configuration
model algorithm. If directed is False, then deg_sequence
is interpreted as the degree sequence of an undirected graph and
in_deg_sequence is ignored. Otherwise, if directed is
True, then deg_sequence is interpreted as the out-degree
sequence and in_deg_sequence as the in-degree sequence of a
directed graph.

.. note::

deg_sequence and in_deg_sequence need not be the same
length.

seed is a random.Random or numpy.random.RandomState instance

This function returns a graph, directed if and only if directed
is True, generated according to the configuration model
:func:configuration_model or :func:directed_configuration_model
functions.

"""
n = len(deg_sequence)
G = nx.empty_graph(n, create_using)
# If empty, return the null graph immediately.
if n == 0:
return G
# Build a list of available degree-repeated nodes.  For example,
# for degree sequence [3, 2, 1, 1, 1], the "stub list" is
# initially [0, 0, 0, 1, 1, 2, 3, 4], that is, node 0 has degree
# 3 and thus is repeated 3 times, etc.
#
# Also, shuffle the stub list in order to get a random sequence of
# node pairs.
if directed:
pairs = zip_longest(deg_sequence, in_deg_sequence, fillvalue=0)
# Unzip the list of pairs into a pair of lists.
out_deg, in_deg = zip(*pairs)

out_stublist = _to_stublist(out_deg)
in_stublist = _to_stublist(in_deg)

seed.shuffle(out_stublist)
seed.shuffle(in_stublist)
else:
stublist = _to_stublist(deg_sequence)
# Choose a random balanced bipartition of the stublist, which
# gives a random pairing of nodes. In this implementation, we
# shuffle the list and then split it in half.
n = len(stublist)
half = n // 2
seed.shuffle(stublist)
out_stublist, in_stublist = stublist[:half], stublist[half:]
return G

[docs]@py_random_state(2)
def configuration_model(deg_sequence, create_using=None, seed=None):
"""Returns a random graph with the given degree sequence.

The configuration model generates a random pseudograph (graph with
parallel edges and self loops) by randomly assigning edges to
match the given degree sequence.

Parameters
----------
deg_sequence :  list of nonnegative integers
Each list entry corresponds to the degree of a node.
create_using : NetworkX graph constructor, optional (default MultiGraph)
Graph type to create. If graph instance, then cleared before populated.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : MultiGraph
A graph with the specified degree sequence.
Nodes are labeled starting at 0 with an index
corresponding to the position in deg_sequence.

Raises
------
NetworkXError
If the degree sequence does not have an even sum.

--------
is_graphical

Notes
-----
As described by Newman [1]_.

A non-graphical degree sequence (not realizable by some simple
graph) is allowed since this function returns graphs with self
loops and parallel edges.  An exception is raised if the degree
sequence does not have an even sum.

This configuration model construction process can lead to
duplicate edges and loops.  You can remove the self-loops and
parallel edges (see below) which will likely result in a graph
that doesn't have the exact degree sequence specified.

The density of self-loops and parallel edges tends to decrease as
the number of nodes increases. However, typically the number of
self-loops will approach a Poisson distribution with a nonzero mean,
and similarly for the number of parallel edges.  Consider a node
with *k* stubs. The probability of being joined to another stub of
the same node is basically (*k* - *1*) / *N*, where *k* is the
degree and *N* is the number of nodes. So the probability of a
self-loop scales like *c* / *N* for some constant *c*. As *N* grows,
this means we expect *c* self-loops. Similarly for parallel edges.

References
----------
.. [1] M.E.J. Newman, "The structure and function of complex networks",
SIAM REVIEW 45-2, pp 167-256, 2003.

Examples
--------
You can create a degree sequence following a particular distribution
by using the one of the distribution functions in
:mod:~networkx.utils.random_sequence (or one of your own). For
example, to create an undirected multigraph on one hundred nodes
with degree sequence chosen from the power law distribution:

>>> sequence = nx.random_powerlaw_tree_sequence(100, tries=5000)
>>> G = nx.configuration_model(sequence)
>>> len(G)
100
>>> actual_degrees = [d for v, d in G.degree()]
>>> actual_degrees == sequence
True

The returned graph is a multigraph, which may have parallel
edges. To remove any parallel edges from the returned graph:

>>> G = nx.Graph(G)

Similarly, to remove self-loops:

>>> G.remove_edges_from(nx.selfloop_edges(G))

"""
if sum(deg_sequence) % 2 != 0:
msg = "Invalid degree sequence: sum of degrees must be even, not odd"
raise nx.NetworkXError(msg)

G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
if G.is_directed():
raise nx.NetworkXNotImplemented("not implemented for directed graphs")

G = _configuration_model(deg_sequence, G, seed=seed)

return G

[docs]@py_random_state(3)
def directed_configuration_model(
in_degree_sequence, out_degree_sequence, create_using=None, seed=None
):
"""Returns a directed_random graph with the given degree sequences.

The configuration model generates a random directed pseudograph
(graph with parallel edges and self loops) by randomly assigning
edges to match the given degree sequences.

Parameters
----------
in_degree_sequence :  list of nonnegative integers
Each list entry corresponds to the in-degree of a node.
out_degree_sequence :  list of nonnegative integers
Each list entry corresponds to the out-degree of a node.
create_using : NetworkX graph constructor, optional (default MultiDiGraph)
Graph type to create. If graph instance, then cleared before populated.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
G : MultiDiGraph
A graph with the specified degree sequences.
Nodes are labeled starting at 0 with an index
corresponding to the position in deg_sequence.

Raises
------
NetworkXError
If the degree sequences do not have the same sum.

--------
configuration_model

Notes
-----
Algorithm as described by Newman [1]_.

A non-graphical degree sequence (not realizable by some simple
graph) is allowed since this function returns graphs with self
loops and parallel edges.  An exception is raised if the degree
sequences does not have the same sum.

This configuration model construction process can lead to
duplicate edges and loops.  You can remove the self-loops and
parallel edges (see below) which will likely result in a graph
that doesn't have the exact degree sequence specified.  This
"finite-size effect" decreases as the size of the graph increases.

References
----------
.. [1] Newman, M. E. J. and Strogatz, S. H. and Watts, D. J.
Random graphs with arbitrary degree distributions and their applications
Phys. Rev. E, 64, 026118 (2001)

Examples
--------
One can modify the in- and out-degree sequences from an existing
directed graph in order to create a new directed graph. For example,
here we modify the directed path graph:

>>> D = nx.DiGraph([(0, 1), (1, 2), (2, 3)])
>>> din = list(d for n, d in D.in_degree())
>>> dout = list(d for n, d in D.out_degree())
>>> din.append(1)
>>> dout[0] = 2
>>> # We now expect an edge from node 0 to a new node, node 3.
... D = nx.directed_configuration_model(din, dout)

The returned graph is a directed multigraph, which may have parallel
edges. To remove any parallel edges from the returned graph:

>>> D = nx.DiGraph(D)

Similarly, to remove self-loops:

>>> D.remove_edges_from(nx.selfloop_edges(D))

"""
if sum(in_degree_sequence) != sum(out_degree_sequence):
msg = "Invalid degree sequences: sequences must have equal sums"
raise nx.NetworkXError(msg)

if create_using is None:
create_using = nx.MultiDiGraph

G = _configuration_model(
out_degree_sequence,
create_using,
directed=True,
in_deg_sequence=in_degree_sequence,
seed=seed,
)

name = "directed configuration_model {} nodes {} edges"
return G

[docs]@py_random_state(1)
def expected_degree_graph(w, seed=None, selfloops=True):
r"""Returns a random graph with given expected degrees.

Given a sequence of expected degrees $W=(w_0,w_1,\ldots,w_{n-1})$
of length $n$ this algorithm assigns an edge between node $u$ and
node $v$ with probability

.. math::

p_{uv} = \frac{w_u w_v}{\sum_k w_k} .

Parameters
----------
w : list
The list of expected degrees.
selfloops: bool (default=True)
Set to False to remove the possibility of self-loop edges.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
Graph

Examples
--------
>>> z = [10 for i in range(100)]
>>> G = nx.expected_degree_graph(z)

Notes
-----
The nodes have integer labels corresponding to index of expected degrees
input sequence.

The complexity of this algorithm is $\mathcal{O}(n+m)$ where $n$ is the
number of nodes and $m$ is the expected number of edges.

The model in [1]_ includes the possibility of self-loop edges.
Set selfloops=False to produce a graph without self loops.

For finite graphs this model doesn't produce exactly the given
expected degree sequence.  Instead the expected degrees are as
follows.

For the case without self loops (selfloops=False),

.. math::

E[deg(u)] = \sum_{v \ne u} p_{uv}
= w_u \left( 1 - \frac{w_u}{\sum_k w_k} \right) .

NetworkX uses the standard convention that a self-loop edge counts 2
in the degree of a node, so with self loops (selfloops=True),

.. math::

E[deg(u)] =  \sum_{v \ne u} p_{uv}  + 2 p_{uu}
= w_u \left( 1 + \frac{w_u}{\sum_k w_k} \right) .

References
----------
.. [1] Fan Chung and L. Lu, Connected components in random graphs with
given expected degree sequences, Ann. Combinatorics, 6,
pp. 125-145, 2002.
.. [2] Joel Miller and Aric Hagberg,
Efficient generation of networks with given expected degrees,
in Algorithms and Models for the Web-Graph (WAW 2011),
Alan Frieze, Paul Horn, and Paweł Prałat (Eds), LNCS 6732,
pp. 115-126, 2011.
"""
n = len(w)
G = nx.empty_graph(n)

# If there are no nodes are no edges in the graph, return the empty graph.
if n == 0 or max(w) == 0:
return G

rho = 1 / sum(w)
# Sort the weights in decreasing order. The original order of the
# weights dictates the order of the (integer) node labels, so we
# need to remember the permutation applied in the sorting.
order = sorted(enumerate(w), key=itemgetter(1), reverse=True)
mapping = {c: u for c, (u, v) in enumerate(order)}
seq = [v for u, v in order]
last = n
if not selfloops:
last -= 1
for u in range(last):
v = u
if not selfloops:
v += 1
factor = seq[u] * rho
p = min(seq[v] * factor, 1)
while v < n and p > 0:
if p != 1:
r = seed.random()
v += math.floor(math.log(r, 1 - p))
if v < n:
q = min(seq[v] * factor, 1)
if seed.random() < q / p:
v += 1
p = q
return G

[docs]def havel_hakimi_graph(deg_sequence, create_using=None):
"""Returns a simple graph with given degree sequence constructed
using the Havel-Hakimi algorithm.

Parameters
----------
deg_sequence: list of integers
Each integer corresponds to the degree of a node (need not be sorted).
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Directed graphs are not allowed.

Raises
------
NetworkXException
For a non-graphical degree sequence (i.e. one
not realizable by some simple graph).

Notes
-----
The Havel-Hakimi algorithm constructs a simple graph by
successively connecting the node of highest degree to other nodes
of highest degree, resorting remaining nodes by degree, and
repeating the process. The resulting graph has a high
degree-associativity.  Nodes are labeled 1,.., len(deg_sequence),
corresponding to their position in deg_sequence.

The basic algorithm is from Hakimi [1]_ and was generalized by
Kleitman and Wang [2]_.

References
----------
.. [1] Hakimi S., On Realizability of a Set of Integers as
Degrees of the Vertices of a Linear Graph. I,
Journal of SIAM, 10(3), pp. 496-506 (1962)
.. [2] Kleitman D.J. and Wang D.L.
Algorithms for Constructing Graphs and Digraphs with Given Valences
and Factors  Discrete Mathematics, 6(1), pp. 79-88 (1973)
"""
if not nx.is_graphical(deg_sequence):
raise nx.NetworkXError("Invalid degree sequence")

p = len(deg_sequence)
G = nx.empty_graph(p, create_using)
if G.is_directed():
raise nx.NetworkXError("Directed graphs are not supported")
num_degs = [[] for i in range(p)]
dmax, dsum, n = 0, 0, 0
for d in deg_sequence:
# Process only the non-zero integers
if d > 0:
num_degs[d].append(n)
dmax, dsum, n = max(dmax, d), dsum + d, n + 1
# Return graph if no edges
if n == 0:
return G

modstubs = [(0, 0)] * (dmax + 1)
# Successively reduce degree sequence by removing the maximum degree
while n > 0:
# Retrieve the maximum degree in the sequence
while len(num_degs[dmax]) == 0:
dmax -= 1
# If there are not enough stubs to connect to, then the sequence is
# not graphical
if dmax > n - 1:
raise nx.NetworkXError("Non-graphical integer sequence")

# Remove largest stub in list
source = num_degs[dmax].pop()
n -= 1
# Reduce the next dmax largest stubs
mslen = 0
k = dmax
for i in range(dmax):
while len(num_degs[k]) == 0:
k -= 1
target = num_degs[k].pop()
n -= 1
if k > 1:
modstubs[mslen] = (k - 1, target)
mslen += 1
# Add back to the list any nonzero stubs that were removed
for i in range(mslen):
(stubval, stubtarget) = modstubs[i]
num_degs[stubval].append(stubtarget)
n += 1

return G

[docs]def directed_havel_hakimi_graph(in_deg_sequence, out_deg_sequence, create_using=None):
"""Returns a directed graph with the given degree sequences.

Parameters
----------
in_deg_sequence :  list of integers
Each list entry corresponds to the in-degree of a node.
out_deg_sequence : list of integers
Each list entry corresponds to the out-degree of a node.
create_using : NetworkX graph constructor, optional (default DiGraph)
Graph type to create. If graph instance, then cleared before populated.

Returns
-------
G : DiGraph
A graph with the specified degree sequences.
Nodes are labeled starting at 0 with an index
corresponding to the position in deg_sequence

Raises
------
NetworkXError
If the degree sequences are not digraphical.

--------
configuration_model

Notes
-----
Algorithm as described by Kleitman and Wang [1]_.

References
----------
.. [1] D.J. Kleitman and D.L. Wang
Algorithms for Constructing Graphs and Digraphs with Given Valences
and Factors Discrete Mathematics, 6(1), pp. 79-88 (1973)
"""
in_deg_sequence = nx.utils.make_list_of_ints(in_deg_sequence)
out_deg_sequence = nx.utils.make_list_of_ints(out_deg_sequence)

# Process the sequences and form two heaps to store degree pairs with
# either zero or nonzero out degrees
sumin, sumout = 0, 0
nin, nout = len(in_deg_sequence), len(out_deg_sequence)
maxn = max(nin, nout)
G = nx.empty_graph(maxn, create_using, default=nx.DiGraph)
if maxn == 0:
return G
maxin = 0
stubheap, zeroheap = [], []
for n in range(maxn):
in_deg, out_deg = 0, 0
if n < nout:
out_deg = out_deg_sequence[n]
if n < nin:
in_deg = in_deg_sequence[n]
if in_deg < 0 or out_deg < 0:
raise nx.NetworkXError(
"Invalid degree sequences. Sequence values must be positive."
)
sumin, sumout, maxin = sumin + in_deg, sumout + out_deg, max(maxin, in_deg)
if in_deg > 0:
stubheap.append((-1 * out_deg, -1 * in_deg, n))
elif out_deg > 0:
zeroheap.append((-1 * out_deg, n))
if sumin != sumout:
raise nx.NetworkXError(
"Invalid degree sequences. Sequences must have equal sums."
)
heapq.heapify(stubheap)
heapq.heapify(zeroheap)

modstubs = [(0, 0, 0)] * (maxin + 1)
# Successively reduce degree sequence by removing the maximum
while stubheap:
# Remove first value in the sequence with a non-zero in degree
(freeout, freein, target) = heapq.heappop(stubheap)
freein *= -1
if freein > len(stubheap) + len(zeroheap):
raise nx.NetworkXError("Non-digraphical integer sequence")

# Attach arcs from the nodes with the most stubs
mslen = 0
for i in range(freein):
if zeroheap and (not stubheap or stubheap[0][0] > zeroheap[0][0]):
(stubout, stubsource) = heapq.heappop(zeroheap)
stubin = 0
else:
(stubout, stubin, stubsource) = heapq.heappop(stubheap)
if stubout == 0:
raise nx.NetworkXError("Non-digraphical integer sequence")
# Check if source is now totally connected
if stubout + 1 < 0 or stubin < 0:
modstubs[mslen] = (stubout + 1, stubin, stubsource)
mslen += 1

# Add the nodes back to the heaps that still have available stubs
for i in range(mslen):
stub = modstubs[i]
if stub[1] < 0:
heapq.heappush(stubheap, stub)
else:
heapq.heappush(zeroheap, (stub[0], stub[2]))
if freeout < 0:
heapq.heappush(zeroheap, (freeout, target))

return G

[docs]def degree_sequence_tree(deg_sequence, create_using=None):
"""Make a tree for the given degree sequence.

A tree has #nodes-#edges=1 so
the degree sequence must have
len(deg_sequence)-sum(deg_sequence)/2=1
"""
# The sum of the degree sequence must be even (for any undirected graph).
degree_sum = sum(deg_sequence)
if degree_sum % 2 != 0:
msg = "Invalid degree sequence: sum of degrees must be even, not odd"
raise nx.NetworkXError(msg)
if len(deg_sequence) - degree_sum // 2 != 1:
msg = (
"Invalid degree sequence: tree must have number of nodes equal"
" to one less than the number of edges"
)
raise nx.NetworkXError(msg)
G = nx.empty_graph(0, create_using)
if G.is_directed():
raise nx.NetworkXError("Directed Graph not supported")

# Sort all degrees greater than 1 in decreasing order.
#
# TODO Does this need to be sorted in reverse order?
deg = sorted((s for s in deg_sequence if s > 1), reverse=True)

# make path graph as backbone
n = len(deg) + 2
last = n

for source in range(1, n - 1):
nedges = deg.pop() - 2
for target in range(last, last + nedges):
last += nedges

# in case we added one too many
if len(G) > len(deg_sequence):
G.remove_node(0)
return G

[docs]@py_random_state(1)
def random_degree_sequence_graph(sequence, seed=None, tries=10):
r"""Returns a simple random graph with the given degree sequence.

If the maximum degree $d_m$ in the sequence is $O(m^{1/4})$ then the
algorithm produces almost uniform random graphs in $O(m d_m)$ time
where $m$ is the number of edges.

Parameters
----------
sequence :  list of integers
Sequence of degrees
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.
tries : int, optional
Maximum number of tries to create a graph

Returns
-------
G : Graph
A graph with the specified degree sequence.
Nodes are labeled starting at 0 with an index
corresponding to the position in the sequence.

Raises
------
NetworkXUnfeasible
If the degree sequence is not graphical.
NetworkXError
If a graph is not produced in specified number of tries

--------
is_graphical, configuration_model

Notes
-----
The generator algorithm [1]_ is not guaranteed to produce a graph.

References
----------
.. [1] Moshen Bayati, Jeong Han Kim, and Amin Saberi,
A sequential algorithm for generating random graphs.
Algorithmica, Volume 58, Number 4, 860-910,
DOI: 10.1007/s00453-009-9340-1

Examples
--------
>>> sequence = [1, 2, 2, 3]
>>> G = nx.random_degree_sequence_graph(sequence, seed=42)
>>> sorted(d for n, d in G.degree())
[1, 2, 2, 3]
"""
DSRG = DegreeSequenceRandomGraph(sequence, seed)
for try_n in range(tries):
try:
return DSRG.generate()
except nx.NetworkXUnfeasible:
pass
raise nx.NetworkXError(f"failed to generate graph in {tries} tries")

class DegreeSequenceRandomGraph:
# class to generate random graphs with a given degree sequence
# use random_degree_sequence_graph()
def __init__(self, degree, rng):
if not nx.is_graphical(degree):
raise nx.NetworkXUnfeasible("degree sequence is not graphical")
self.rng = rng
self.degree = list(degree)
# node labels are integers 0,...,n-1
self.m = sum(self.degree) / 2.0  # number of edges
try:
self.dmax = max(self.degree)  # maximum degree
except ValueError:
self.dmax = 0

def generate(self):
# remaining_degree is mapping from int->remaining degree
self.remaining_degree = dict(enumerate(self.degree))
# add all nodes to make sure we get isolated nodes
self.graph = nx.Graph()
# remove zero degree nodes
for n, d in list(self.remaining_degree.items()):
if d == 0:
del self.remaining_degree[n]
if len(self.remaining_degree) > 0:
# build graph in three phases according to how many unmatched edges
self.phase1()
self.phase2()
self.phase3()
return self.graph

def update_remaining(self, u, v, aux_graph=None):
# decrement remaining nodes, modify auxiliary graph if in phase3
if aux_graph is not None:
# remove edges from auxiliary graph
aux_graph.remove_edge(u, v)
if self.remaining_degree[u] == 1:
del self.remaining_degree[u]
if aux_graph is not None:
aux_graph.remove_node(u)
else:
self.remaining_degree[u] -= 1
if self.remaining_degree[v] == 1:
del self.remaining_degree[v]
if aux_graph is not None:
aux_graph.remove_node(v)
else:
self.remaining_degree[v] -= 1

def p(self, u, v):
# degree probability
return 1 - self.degree[u] * self.degree[v] / (4.0 * self.m)

def q(self, u, v):
# remaining degree probability
norm = max(self.remaining_degree.values()) ** 2
return self.remaining_degree[u] * self.remaining_degree[v] / norm

def suitable_edge(self):
"""Returns True if and only if an arbitrary remaining node can
potentially be joined with some other remaining node.

"""
nodes = iter(self.remaining_degree)
u = next(nodes)
return any(v not in self.graph[u] for v in nodes)

def phase1(self):
# choose node pairs from (degree) weighted distribution
rem_deg = self.remaining_degree
while sum(rem_deg.values()) >= 2 * self.dmax**2:
u, v = sorted(random_weighted_sample(rem_deg, 2, self.rng))
if self.graph.has_edge(u, v):
continue
if self.rng.random() < self.p(u, v):  # accept edge
self.update_remaining(u, v)

def phase2(self):
# choose remaining nodes uniformly at random and use rejection sampling
remaining_deg = self.remaining_degree
rng = self.rng
while len(remaining_deg) >= 2 * self.dmax:
while True:
u, v = sorted(rng.sample(list(remaining_deg.keys()), 2))
if self.graph.has_edge(u, v):
continue
if rng.random() < self.q(u, v):
break
if rng.random() < self.p(u, v):  # accept edge
self.update_remaining(u, v)

def phase3(self):
# build potential remaining edges and choose with rejection sampling
potential_edges = combinations(self.remaining_degree, 2)
# build auxiliary graph of potential edges not already in graph
H = nx.Graph(
[(u, v) for (u, v) in potential_edges if not self.graph.has_edge(u, v)]
)
rng = self.rng
while self.remaining_degree:
if not self.suitable_edge():
raise nx.NetworkXUnfeasible("no suitable edges left")
while True:
u, v = sorted(rng.choice(list(H.edges())))
if rng.random() < self.q(u, v):
break
if rng.random() < self.p(u, v):  # accept edge