# -*- coding: utf-8 -*-
"""Generate graphs with a given degree sequence or expected degree sequence.
"""
# Copyright (C) 2004-2011 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
import heapq
from itertools import combinations, permutations
import math
from operator import itemgetter
import random
import networkx as nx
from networkx.utils import random_weighted_sample
__author__ = "\n".join(['Aric Hagberg <aric.hagberg@gmail.com>',
'Pieter Swart <swart@lanl.gov>',
'Dan Schult <dschult@colgate.edu>'
'Joel Miller <joel.c.miller.research@gmail.com>',
'Nathan Lemons <nlemons@gmail.com>'])
__all__ = ['configuration_model',
'directed_configuration_model',
'expected_degree_graph',
'havel_hakimi_graph',
'degree_sequence_tree',
'random_degree_sequence_graph']
[docs]def configuration_model(deg_sequence,create_using=None,seed=None):
"""Return 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 integers
Each list entry corresponds to the degree of a node.
create_using : graph, optional (default MultiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
Seed for random number generator.
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.
See Also
--------
is_valid_degree_sequence
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. This
"finite-size effect" decreases as the size of the graph increases.
References
----------
.. [1] M.E.J. Newman, "The structure and function of complex networks",
SIAM REVIEW 45-2, pp 167-256, 2003.
Examples
--------
>>> from networkx.utils import powerlaw_sequence
>>> z=nx.utils.create_degree_sequence(100,powerlaw_sequence)
>>> G=nx.configuration_model(z)
To remove parallel edges:
>>> G=nx.Graph(G)
To remove self loops:
>>> G.remove_edges_from(G.selfloop_edges())
"""
if not sum(deg_sequence)%2 ==0:
raise nx.NetworkXError('Invalid degree sequence')
if create_using is None:
create_using = nx.MultiGraph()
elif create_using.is_directed():
raise nx.NetworkXError("Directed Graph not supported")
if not seed is None:
random.seed(seed)
# start with empty N-node graph
N=len(deg_sequence)
# allow multiedges and selfloops
G=nx.empty_graph(N,create_using)
if N==0 or max(deg_sequence)==0: # done if no edges
return G
# build stublist, a list of available degree-repeated stubs
# e.g. for deg_sequence=[3,2,1,1,1]
# initially, stublist=[1,1,1,2,2,3,4,5]
# i.e., node 1 has degree=3 and is repeated 3 times, etc.
stublist=[]
for n in G:
for i in range(deg_sequence[n]):
stublist.append(n)
# shuffle stublist and assign pairs by removing 2 elements at a time
random.shuffle(stublist)
while stublist:
n1 = stublist.pop()
n2 = stublist.pop()
G.add_edge(n1,n2)
G.name="configuration_model %d nodes %d edges"%(G.order(),G.size())
return G
[docs]def directed_configuration_model(in_degree_sequence,
out_degree_sequence,
create_using=None,seed=None):
"""Return 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 integers
Each list entry corresponds to the in-degree of a node.
out_degree_sequence : list of integers
Each list entry corresponds to the out-degree of a node.
create_using : graph, optional (default MultiDiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
Seed for random number generator.
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.
See Also
--------
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
--------
>>> D=nx.DiGraph([(0,1),(1,2),(2,3)]) # directed path graph
>>> din=list(D.in_degree().values())
>>> dout=list(D.out_degree().values())
>>> din.append(1)
>>> dout[0]=2
>>> D=nx.directed_configuration_model(din,dout)
To remove parallel edges:
>>> D=nx.DiGraph(D)
To remove self loops:
>>> D.remove_edges_from(D.selfloop_edges())
"""
if not sum(in_degree_sequence) == sum(out_degree_sequence):
raise nx.NetworkXError('Invalid degree sequences. '
'Sequences must have equal sums.')
if create_using is None:
create_using = nx.MultiDiGraph()
if not seed is None:
random.seed(seed)
nin=len(in_degree_sequence)
nout=len(out_degree_sequence)
# pad in- or out-degree sequence with zeros to match lengths
if nin>nout:
out_degree_sequence.extend((nin-nout)*[0])
else:
in_degree_sequence.extend((nout-nin)*[0])
# start with empty N-node graph
N=len(in_degree_sequence)
# allow multiedges and selfloops
G=nx.empty_graph(N,create_using)
if N==0 or max(in_degree_sequence)==0: # done if no edges
return G
# build stublists of available degree-repeated stubs
# e.g. for degree_sequence=[3,2,1,1,1]
# initially, stublist=[1,1,1,2,2,3,4,5]
# i.e., node 1 has degree=3 and is repeated 3 times, etc.
in_stublist=[]
for n in G:
for i in range(in_degree_sequence[n]):
in_stublist.append(n)
out_stublist=[]
for n in G:
for i in range(out_degree_sequence[n]):
out_stublist.append(n)
# shuffle stublists and assign pairs by removing 2 elements at a time
random.shuffle(in_stublist)
random.shuffle(out_stublist)
while in_stublist and out_stublist:
source = out_stublist.pop()
target = in_stublist.pop()
G.add_edge(source,target)
G.name="directed configuration_model %d nodes %d edges"%(G.order(),G.size())
return G
[docs]def expected_degree_graph(w, seed=None, selfloops=True):
r"""Return 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 : hashable object, optional
The seed for the random number generator.
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 n==0 or max(w)==0: # done if no edges
return G
if seed is not None:
random.seed(seed)
rho = 1/float(sum(w))
# sort weights, largest first
# preserve order of weights for integer node label mapping
order = sorted(enumerate(w),key=itemgetter(1),reverse=True)
mapping = dict((c,uv[0]) for c,uv 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 = seq[v]*factor
if p>1:
p = 1
while v<n and p>0:
if p != 1:
r = random.random()
v += int(math.floor(math.log(r)/math.log(1-p)))
if v < n:
q = seq[v]*factor
if q>1:
q = 1
if random.random() < q/p:
G.add_edge(mapping[u],mapping[v])
v += 1
p = q
return G
[docs]def havel_hakimi_graph(deg_sequence,create_using=None):
"""Return 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 : graph, optional (default Graph)
Return graph of this type. The instance will be cleared.
Multigraphs and 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.
See Theorem 1.4 in [1]_.
This algorithm is also used in the function is_valid_degree_sequence.
References
----------
.. [1] G. Chartrand and L. Lesniak, "Graphs and Digraphs",
Chapman and Hall/CRC, 1996.
"""
if not nx.is_valid_degree_sequence(deg_sequence):
raise nx.NetworkXError('Invalid degree sequence')
if create_using is not None:
if create_using.is_directed():
raise nx.NetworkXError("Directed Graph not supported")
if create_using.is_multigraph():
raise nx.NetworkXError("Havel-Hakimi requires simple graph")
N=len(deg_sequence)
G=nx.empty_graph(N,create_using)
if N==0 or max(deg_sequence)==0: # done if no edges
return G
# form list of [stubs,name] for each node.
stublist=[ [deg_sequence[n],n] for n in G]
# Now connect the stubs
while stublist:
stublist.sort()
if stublist[0][0]<0: # took too many off some vertex
return False # should not happen if deg_seq is valid
(freestubs,source) = stublist.pop() # the node with the most stubs
if freestubs==0: break # the rest must also be 0 --Done!
if freestubs > len(stublist): # Trouble--can't make that many edges
return False # should not happen if deg_seq is valid
# attach edges to biggest nodes
for stubtarget in stublist[-freestubs:]:
G.add_edge(source, stubtarget[1])
stubtarget[0] -= 1 # updating stublist on the fly
G.name="havel_hakimi_graph %d nodes %d edges"%(G.order(),G.size())
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
"""
if not len(deg_sequence)-sum(deg_sequence)/2.0 == 1.0:
raise nx.NetworkXError("Degree sequence invalid")
if create_using is not None and create_using.is_directed():
raise nx.NetworkXError("Directed Graph not supported")
# single node tree
if len(deg_sequence)==1:
G=nx.empty_graph(0,create_using)
return G
# all degrees greater than 1
deg=[s for s in deg_sequence if s>1]
deg.sort(reverse=True)
# make path graph as backbone
n=len(deg)+2
G=nx.path_graph(n,create_using)
last=n
# add the leaves
for source in range(1,n-1):
nedges=deg.pop()-2
for target in range(last,last+nedges):
G.add_edge(source, target)
last+=nedges
# in case we added one too many
if len(G.degree())>len(deg_sequence):
G.remove_node(0)
return G
[docs]def random_degree_sequence_graph(sequence, seed=None, tries=10):
r"""Return 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 : hashable object, optional
Seed for random number generator
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
See Also
--------
is_valid_degree_sequence, 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)
>>> sorted(G.degree().values())
[1, 2, 2, 3]
"""
DSRG = DegreeSequenceRandomGraph(sequence, seed=seed)
for try_n in range(tries):
try:
return DSRG.generate()
except nx.NetworkXUnfeasible:
pass
raise nx.NetworkXError('failed to generate graph in %d tries'%tries)
class DegreeSequenceRandomGraph(object):
# class to generate random graphs with a given degree sequence
# use random_degree_sequence_graph()
def __init__(self, degree, seed=None):
if not nx.is_valid_degree_sequence(degree):
raise nx.NetworkXUnfeasible('degree sequence is not graphical')
if seed is not None:
random.seed(seed)
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()
self.graph.add_nodes_from(self.remaining_degree)
# 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 auxilliary graph if in phase3
if aux_graph is not None:
# remove edges from auxilliary 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 = float(max(self.remaining_degree.values()))**2
return self.remaining_degree[u]*self.remaining_degree[v]/norm
def suitable_edge(self):
# Check if there is a suitable edge that is not in the graph
# True if an (arbitrary) remaining node has at least one possible
# connection to another remaining node
nodes = iter(self.remaining_degree)
u = next(nodes) # one arbitrary node
for v in nodes: # loop over all other remaining nodes
if not self.graph.has_edge(u, v):
return True
return False
def phase1(self):
# choose node pairs from (degree) weighted distribution
while sum(self.remaining_degree.values()) >= 2 * self.dmax**2:
u,v = sorted(random_weighted_sample(self.remaining_degree, 2))
if self.graph.has_edge(u,v):
continue
if random.random() < self.p(u,v): # accept edge
self.graph.add_edge(u,v)
self.update_remaining(u,v)
def phase2(self):
# choose remaining nodes uniformly at random and use rejection sampling
while len(self.remaining_degree) >= 2 * self.dmax:
norm = float(max(self.remaining_degree.values()))**2
while True:
u,v = sorted(random.sample(self.remaining_degree.keys(), 2))
if self.graph.has_edge(u,v):
continue
if random.random() < self.q(u,v):
break
if random.random() < self.p(u,v): # accept edge
self.graph.add_edge(u,v)
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 auxilliary 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)])
while self.remaining_degree:
if not self.suitable_edge():
raise nx.NetworkXUnfeasible('no suitable edges left')
while True:
u,v = sorted(random.choice(H.edges()))
if random.random() < self.q(u,v):
break
if random.random() < self.p(u,v): # accept edge
self.graph.add_edge(u,v)
self.update_remaining(u,v, aux_graph=H)