"""
Generators for some directed graphs, including growing network (GN) graphs and
scale-free graphs.
"""
import numbers
from collections import Counter
import networkx as nx
from networkx.generators.classic import empty_graph
from networkx.utils import (
discrete_sequence,
np_random_state,
py_random_state,
weighted_choice,
)
__all__ = [
"gn_graph",
"gnc_graph",
"gnr_graph",
"random_k_out_graph",
"scale_free_graph",
]
[docs]
@py_random_state(3)
@nx._dispatchable(graphs=None, returns_graph=True)
def gn_graph(n, kernel=None, create_using=None, seed=None):
"""Returns the growing network (GN) digraph with `n` nodes.
The GN graph is built by adding nodes one at a time with a link to one
previously added node. The target node for the link is chosen with
probability based on degree. The default attachment kernel is a linear
function of the degree of a node.
The graph is always a (directed) tree.
Parameters
----------
n : int
The number of nodes for the generated graph.
kernel : function
The attachment kernel.
create_using : NetworkX graph constructor, optional (default DiGraph)
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>`.
Examples
--------
To create the undirected GN graph, use the :meth:`~DiGraph.to_directed`
method::
>>> D = nx.gn_graph(10) # the GN graph
>>> G = D.to_undirected() # the undirected version
To specify an attachment kernel, use the `kernel` keyword argument::
>>> D = nx.gn_graph(10, kernel=lambda x: x**1.5) # A_k = k^1.5
References
----------
.. [1] P. L. Krapivsky and S. Redner,
Organization of Growing Random Networks,
Phys. Rev. E, 63, 066123, 2001.
"""
G = empty_graph(1, create_using, default=nx.DiGraph)
if not G.is_directed():
raise nx.NetworkXError("create_using must indicate a Directed Graph")
if kernel is None:
def kernel(x):
return x
if n == 1:
return G
G.add_edge(1, 0) # get started
ds = [1, 1] # degree sequence
for source in range(2, n):
# compute distribution from kernel and degree
dist = [kernel(d) for d in ds]
# choose target from discrete distribution
target = discrete_sequence(1, distribution=dist, seed=seed)[0]
G.add_edge(source, target)
ds.append(1) # the source has only one link (degree one)
ds[target] += 1 # add one to the target link degree
return G
[docs]
@py_random_state(3)
@nx._dispatchable(graphs=None, returns_graph=True)
def gnr_graph(n, p, create_using=None, seed=None):
"""Returns the growing network with redirection (GNR) digraph with `n`
nodes and redirection probability `p`.
The GNR graph is built by adding nodes one at a time with a link to one
previously added node. The previous target node is chosen uniformly at
random. With probability `p` the link is instead "redirected" to the
successor node of the target.
The graph is always a (directed) tree.
Parameters
----------
n : int
The number of nodes for the generated graph.
p : float
The redirection probability.
create_using : NetworkX graph constructor, optional (default DiGraph)
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>`.
Examples
--------
To create the undirected GNR graph, use the :meth:`~DiGraph.to_directed`
method::
>>> D = nx.gnr_graph(10, 0.5) # the GNR graph
>>> G = D.to_undirected() # the undirected version
References
----------
.. [1] P. L. Krapivsky and S. Redner,
Organization of Growing Random Networks,
Phys. Rev. E, 63, 066123, 2001.
"""
G = empty_graph(1, create_using, default=nx.DiGraph)
if not G.is_directed():
raise nx.NetworkXError("create_using must indicate a Directed Graph")
if n == 1:
return G
for source in range(1, n):
target = seed.randrange(0, source)
if seed.random() < p and target != 0:
target = next(G.successors(target))
G.add_edge(source, target)
return G
[docs]
@py_random_state(2)
@nx._dispatchable(graphs=None, returns_graph=True)
def gnc_graph(n, create_using=None, seed=None):
"""Returns the growing network with copying (GNC) digraph with `n` nodes.
The GNC graph is built by adding nodes one at a time with a link to one
previously added node (chosen uniformly at random) and to all of that
node's successors.
Parameters
----------
n : int
The number of nodes for the generated graph.
create_using : NetworkX graph constructor, optional (default DiGraph)
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>`.
References
----------
.. [1] P. L. Krapivsky and S. Redner,
Network Growth by Copying,
Phys. Rev. E, 71, 036118, 2005k.},
"""
G = empty_graph(1, create_using, default=nx.DiGraph)
if not G.is_directed():
raise nx.NetworkXError("create_using must indicate a Directed Graph")
if n == 1:
return G
for source in range(1, n):
target = seed.randrange(0, source)
for succ in G.successors(target):
G.add_edge(source, succ)
G.add_edge(source, target)
return G
[docs]
@py_random_state(6)
@nx._dispatchable(graphs=None, returns_graph=True)
def scale_free_graph(
n,
alpha=0.41,
beta=0.54,
gamma=0.05,
delta_in=0.2,
delta_out=0,
seed=None,
initial_graph=None,
):
"""Returns a scale-free directed graph.
Parameters
----------
n : integer
Number of nodes in graph
alpha : float
Probability for adding a new node connected to an existing node
chosen randomly according to the in-degree distribution.
beta : float
Probability for adding an edge between two existing nodes.
One existing node is chosen randomly according the in-degree
distribution and the other chosen randomly according to the out-degree
distribution.
gamma : float
Probability for adding a new node connected to an existing node
chosen randomly according to the out-degree distribution.
delta_in : float
Bias for choosing nodes from in-degree distribution.
delta_out : float
Bias for choosing nodes from out-degree distribution.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : MultiDiGraph instance, optional
Build the scale-free graph starting from this initial MultiDiGraph,
if provided.
Returns
-------
MultiDiGraph
Examples
--------
Create a scale-free graph on one hundred nodes::
>>> G = nx.scale_free_graph(100)
Notes
-----
The sum of `alpha`, `beta`, and `gamma` must be 1.
References
----------
.. [1] B. Bollobás, C. Borgs, J. Chayes, and O. Riordan,
Directed scale-free graphs,
Proceedings of the fourteenth annual ACM-SIAM Symposium on
Discrete Algorithms, 132--139, 2003.
"""
def _choose_node(candidates, node_list, delta):
if delta > 0:
bias_sum = len(node_list) * delta
p_delta = bias_sum / (bias_sum + len(candidates))
if seed.random() < p_delta:
return seed.choice(node_list)
return seed.choice(candidates)
if initial_graph is not None and hasattr(initial_graph, "_adj"):
if not isinstance(initial_graph, nx.MultiDiGraph):
raise nx.NetworkXError("initial_graph must be a MultiDiGraph.")
G = initial_graph
else:
# Start with 3-cycle
G = nx.MultiDiGraph([(0, 1), (1, 2), (2, 0)])
if alpha <= 0:
raise ValueError("alpha must be > 0.")
if beta <= 0:
raise ValueError("beta must be > 0.")
if gamma <= 0:
raise ValueError("gamma must be > 0.")
if abs(alpha + beta + gamma - 1.0) >= 1e-9:
raise ValueError("alpha+beta+gamma must equal 1.")
if delta_in < 0:
raise ValueError("delta_in must be >= 0.")
if delta_out < 0:
raise ValueError("delta_out must be >= 0.")
# pre-populate degree states
vs = sum((count * [idx] for idx, count in G.out_degree()), [])
ws = sum((count * [idx] for idx, count in G.in_degree()), [])
# pre-populate node state
node_list = list(G.nodes())
# see if there already are number-based nodes
numeric_nodes = [n for n in node_list if isinstance(n, numbers.Number)]
if len(numeric_nodes) > 0:
# set cursor for new nodes appropriately
cursor = max(int(n.real) for n in numeric_nodes) + 1
else:
# or start at zero
cursor = 0
while len(G) < n:
r = seed.random()
# random choice in alpha,beta,gamma ranges
if r < alpha:
# alpha
# add new node v
v = cursor
cursor += 1
# also add to node state
node_list.append(v)
# choose w according to in-degree and delta_in
w = _choose_node(ws, node_list, delta_in)
elif r < alpha + beta:
# beta
# choose v according to out-degree and delta_out
v = _choose_node(vs, node_list, delta_out)
# choose w according to in-degree and delta_in
w = _choose_node(ws, node_list, delta_in)
else:
# gamma
# choose v according to out-degree and delta_out
v = _choose_node(vs, node_list, delta_out)
# add new node w
w = cursor
cursor += 1
# also add to node state
node_list.append(w)
# add edge to graph
G.add_edge(v, w)
# update degree states
vs.append(v)
ws.append(w)
return G
@py_random_state(4)
@nx._dispatchable(graphs=None, returns_graph=True)
def random_uniform_k_out_graph(n, k, self_loops=True, with_replacement=True, seed=None):
"""Returns a random `k`-out graph with uniform attachment.
A random `k`-out graph with uniform attachment is a multidigraph
generated by the following algorithm. For each node *u*, choose
`k` nodes *v* uniformly at random (with replacement). Add a
directed edge joining *u* to *v*.
Parameters
----------
n : int
The number of nodes in the returned graph.
k : int
The out-degree of each node in the returned graph.
self_loops : bool
If True, self-loops are allowed when generating the graph.
with_replacement : bool
If True, neighbors are chosen with replacement and the
returned graph will be a directed multigraph. Otherwise,
neighbors are chosen without replacement and the returned graph
will be a directed graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
NetworkX graph
A `k`-out-regular directed graph generated according to the
above algorithm. It will be a multigraph if and only if
`with_replacement` is True.
Raises
------
ValueError
If `with_replacement` is False and `k` is greater than
`n`.
See also
--------
random_k_out_graph
Notes
-----
The return digraph or multidigraph may not be strongly connected, or
even weakly connected.
If `with_replacement` is True, this function is similar to
:func:`random_k_out_graph`, if that function had parameter `alpha`
set to positive infinity.
"""
if with_replacement:
create_using = nx.MultiDiGraph()
def sample(v, nodes):
if not self_loops:
nodes = nodes - {v}
return (seed.choice(list(nodes)) for i in range(k))
else:
create_using = nx.DiGraph()
def sample(v, nodes):
if not self_loops:
nodes = nodes - {v}
return seed.sample(list(nodes), k)
G = nx.empty_graph(n, create_using)
nodes = set(G)
for u in G:
G.add_edges_from((u, v) for v in sample(u, nodes))
return G
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def random_k_out_graph(n, k, alpha, self_loops=True, seed=None):
"""Returns a random `k`-out graph with preferential attachment.
.. versionchanged:: 3.5
Different implementations will be used based on whether NumPy is
available. See Notes for details.
A random `k`-out graph with preferential attachment is a
multidigraph generated by the following algorithm.
1. Begin with an empty digraph, and initially set each node to have
weight `alpha`.
2. Choose a node `u` with out-degree less than `k` uniformly at
random.
3. Choose a node `v` from with probability proportional to its
weight.
4. Add a directed edge from `u` to `v`, and increase the weight
of `v` by one.
5. If each node has out-degree `k`, halt, otherwise repeat from
step 2.
For more information on this model of random graph, see [1]_.
Parameters
----------
n : int
The number of nodes in the returned graph.
k : int
The out-degree of each node in the returned graph.
alpha : float
A positive :class:`float` representing the initial weight of
each vertex. A higher number means that in step 3 above, nodes
will be chosen more like a true uniformly random sample, and a
lower number means that nodes are more likely to be chosen as
their in-degree increases. If this parameter is not positive, a
:exc:`ValueError` is raised.
self_loops : bool
If True, self-loops are allowed when generating the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
:class:`~networkx.classes.MultiDiGraph`
A `k`-out-regular multidigraph generated according to the above
algorithm.
Raises
------
ValueError
If `alpha` is not positive.
Notes
-----
The returned multidigraph may not be strongly connected, or even
weakly connected.
`random_k_out_graph` has two implementations: an array-based formulation that
uses `numpy` (``_random_k_out_graph_numpy``), and a pure-Python
implementation (``_random_k_out_graph_python``).
The NumPy implementation is more performant, especially for large `n`, and is
therefore used by default. If NumPy is not installed in the environment,
then the pure Python implementation is executed.
However, you can explicitly control which implementation is executed by directly
calling the corresponding function::
# Use numpy if available, else Python
nx.random_k_out_graph(1000, 5, alpha=1)
# Use the numpy-based implementation (raises ImportError if numpy not installed)
nx.generators.directed._random_k_out_graph_numpy(1000, 5, alpha=1)
# Use the Python-based implementation
nx.generators.directed._random_k_out_graph_python(1000, 5, alpha=1)
References
----------
.. [1] Peterson, Nicholas R., and Boris Pittel.
"Distance between two random `k`-out digraphs, with and without preferential attachment."
arXiv preprint arXiv:1311.5961 (2013) <https://arxiv.org/abs/1311.5961>.
"""
if alpha < 0:
raise ValueError("alpha must be positive")
try: # Use numpy if available, otherwise fall back to pure Python implementation
return _random_k_out_graph_numpy(n, k, alpha, self_loops, seed)
except ImportError:
return _random_k_out_graph_python(n, k, alpha, self_loops, seed)
@np_random_state(4)
def _random_k_out_graph_numpy(n, k, alpha, self_loops=True, seed=None):
import numpy as np
G = nx.empty_graph(n, create_using=nx.MultiDiGraph)
nodes = np.arange(n)
remaining_mask = np.full(n, True)
weights = np.full(n, alpha)
total_weight = n * alpha
out_strengths = np.zeros(n)
for i in range(k * n):
u = seed.choice(nodes[remaining_mask])
if self_loops:
v = seed.choice(nodes, p=weights / total_weight)
else: # Ignore weight of u when selecting v
u_weight = weights[u]
weights[u] = 0
v = seed.choice(nodes, p=weights / (total_weight - u_weight))
weights[u] = u_weight
G.add_edge(u, v)
weights[v] += 1
total_weight += 1
out_strengths[u] += 1
if out_strengths[u] == k:
remaining_mask[u] = False
return G
@py_random_state(4)
def _random_k_out_graph_python(n, k, alpha, self_loops=True, seed=None):
G = nx.empty_graph(n, create_using=nx.MultiDiGraph)
weights = Counter({v: alpha for v in G})
out_strengths = Counter({v: 0 for v in G})
for i in range(k * n):
u = seed.choice(list(out_strengths.keys()))
# If self-loops are not allowed, make the source node `u` have
# weight zero.
if not self_loops:
uweight = weights.pop(u)
v = weighted_choice(weights, seed=seed)
if not self_loops:
weights[u] = uweight
G.add_edge(u, v)
weights[v] += 1
out_strengths[u] += 1
if out_strengths[u] == k:
out_strengths.pop(u)
return G
