# Copyright(C) 2011-2019 by
# Ben Edwards <bedwards@cs.unm.edu>
# Aric Hagberg <hagberg@lanl.gov>
# Konstantinos Karakatsanis <dinoskarakas@gmail.com>
# All rights reserved.
# BSD license.
#
# Authors: Ben Edwards (bedwards@cs.unm.edu)
# Aric Hagberg (hagberg@lanl.gov)
# Konstantinos Karakatsanis (dinoskarakas@gmail.com)
# Jean-Gabriel Young (jean.gabriel.young@gmail.com)
"""Generators for classes of graphs used in studying social networks."""
from __future__ import division
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']
[docs]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)
See also
--------
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)
G.add_edges_from(edges)
return G
[docs]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
adjacent clique.
Parameters
----------
l : int
number of cliques
k : int
size of cliques
Returns
-------
G : NetworkX Graph
connected 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.connected_caveman_graph(3, 3)
References
----------
.. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
Amer. J. Soc. 105, 493-527, 1999.
"""
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)
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
Probabilty 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)
G.add_edge(u, x)
return G
[docs]@py_random_state(3)
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],.25,.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)
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] or
Examples
--------
>>> G = nx.planted_partition_graph(4, 3, 0.5, 0.1, seed=42)
See Also
--------
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, directed)
[docs]@py_random_state(6)
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
Probabilty 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]
See Also
--------
random_partition_graph
Examples
--------
>>> G = nx.gaussian_random_partition_graph(100,10,10,.25,.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, float(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, directed, seed)
[docs]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)
See Also
--------
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
simply adds the link without removing any link from the cliques.
"""
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)
G.add_edges_from(edges)
G.add_edge(i * clique_size + 1, (i + 1) * clique_size %
(num_cliques * clique_size))
return G
[docs]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)
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
See Also
--------
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(0, 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(0, len(sizes) + 1)]
g.graph['partition'] = [set(nodelist[size_cumsum[x]:size_cumsum[x + 1]])
for x in range(0, len(size_cumsum) - 1)]
# Setup nodes and graph name
for block_id, nodes in enumerate(g.graph['partition']):
for node in nodes:
g.add_node(node, block=block_id)
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]:
g.add_edge(*e)
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:
g.add_edge(*e)
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)
g.add_edge(*e) # __safe
except StopIteration:
break
else:
for e in edges:
if seed.random() < p[i][j]:
g.add_edge(*e) # __safe
return g