"""
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.
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.
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")
return _random_k_out_graph_numpy(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
