502 lines
15 KiB
Python
502 lines
15 KiB
Python
"""
|
|
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, py_random_state, weighted_choice
|
|
|
|
__all__ = [
|
|
"gn_graph",
|
|
"gnc_graph",
|
|
"gnr_graph",
|
|
"random_k_out_graph",
|
|
"scale_free_graph",
|
|
]
|
|
|
|
|
|
@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
|
|
|
|
|
|
@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
|
|
|
|
|
|
@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
|
|
|
|
|
|
@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
|
|
|
|
|
|
@py_random_state(4)
|
|
@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")
|
|
G = nx.empty_graph(n, create_using=nx.MultiDiGraph)
|
|
weights = Counter({v: alpha for v in G})
|
|
for i in range(k * n):
|
|
u = seed.choice([v for v, d in G.out_degree() if d < k])
|
|
# If self-loops are not allowed, make the source node `u` have
|
|
# weight zero.
|
|
if not self_loops:
|
|
adjustment = Counter({u: weights[u]})
|
|
else:
|
|
adjustment = Counter()
|
|
v = weighted_choice(weights - adjustment, seed=seed)
|
|
G.add_edge(u, v)
|
|
weights[v] += 1
|
|
return G
|