494 lines
15 KiB
Python
494 lines
15 KiB
Python
"""
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Generators for some directed graphs, including growing network (GN) graphs and
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scale-free graphs.
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"""
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import numbers
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from collections import Counter
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import networkx as nx
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from networkx.generators.classic import empty_graph
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from networkx.utils import discrete_sequence, py_random_state, weighted_choice
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__all__ = [
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"gn_graph",
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"gnc_graph",
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"gnr_graph",
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"random_k_out_graph",
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"scale_free_graph",
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]
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@py_random_state(3)
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def gn_graph(n, kernel=None, create_using=None, seed=None):
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"""Returns the growing network (GN) digraph with `n` nodes.
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The GN graph is built by adding nodes one at a time with a link to one
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previously added node. The target node for the link is chosen with
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probability based on degree. The default attachment kernel is a linear
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function of the degree of a node.
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The graph is always a (directed) tree.
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Parameters
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----------
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n : int
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The number of nodes for the generated graph.
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kernel : function
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The attachment kernel.
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create_using : NetworkX graph constructor, optional (default DiGraph)
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Graph type to create. If graph instance, then cleared before populated.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Examples
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--------
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To create the undirected GN graph, use the :meth:`~DiGraph.to_directed`
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method::
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>>> D = nx.gn_graph(10) # the GN graph
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>>> G = D.to_undirected() # the undirected version
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To specify an attachment kernel, use the `kernel` keyword argument::
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>>> D = nx.gn_graph(10, kernel=lambda x: x ** 1.5) # A_k = k^1.5
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References
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----------
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.. [1] P. L. Krapivsky and S. Redner,
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Organization of Growing Random Networks,
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Phys. Rev. E, 63, 066123, 2001.
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"""
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G = empty_graph(1, create_using, default=nx.DiGraph)
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if not G.is_directed():
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raise nx.NetworkXError("create_using must indicate a Directed Graph")
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if kernel is None:
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def kernel(x):
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return x
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if n == 1:
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return G
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G.add_edge(1, 0) # get started
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ds = [1, 1] # degree sequence
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for source in range(2, n):
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# compute distribution from kernel and degree
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dist = [kernel(d) for d in ds]
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# choose target from discrete distribution
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target = discrete_sequence(1, distribution=dist, seed=seed)[0]
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G.add_edge(source, target)
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ds.append(1) # the source has only one link (degree one)
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ds[target] += 1 # add one to the target link degree
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return G
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@py_random_state(3)
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def gnr_graph(n, p, create_using=None, seed=None):
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"""Returns the growing network with redirection (GNR) digraph with `n`
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nodes and redirection probability `p`.
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The GNR graph is built by adding nodes one at a time with a link to one
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previously added node. The previous target node is chosen uniformly at
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random. With probabiliy `p` the link is instead "redirected" to the
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successor node of the target.
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The graph is always a (directed) tree.
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Parameters
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----------
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n : int
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The number of nodes for the generated graph.
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p : float
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The redirection probability.
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create_using : NetworkX graph constructor, optional (default DiGraph)
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Graph type to create. If graph instance, then cleared before populated.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Examples
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--------
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To create the undirected GNR graph, use the :meth:`~DiGraph.to_directed`
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method::
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>>> D = nx.gnr_graph(10, 0.5) # the GNR graph
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>>> G = D.to_undirected() # the undirected version
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References
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----------
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.. [1] P. L. Krapivsky and S. Redner,
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Organization of Growing Random Networks,
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Phys. Rev. E, 63, 066123, 2001.
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"""
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G = empty_graph(1, create_using, default=nx.DiGraph)
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if not G.is_directed():
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raise nx.NetworkXError("create_using must indicate a Directed Graph")
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if n == 1:
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return G
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for source in range(1, n):
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target = seed.randrange(0, source)
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if seed.random() < p and target != 0:
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target = next(G.successors(target))
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G.add_edge(source, target)
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return G
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@py_random_state(2)
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def gnc_graph(n, create_using=None, seed=None):
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"""Returns the growing network with copying (GNC) digraph with `n` nodes.
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The GNC graph is built by adding nodes one at a time with a link to one
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previously added node (chosen uniformly at random) and to all of that
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node's successors.
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Parameters
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----------
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n : int
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The number of nodes for the generated graph.
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create_using : NetworkX graph constructor, optional (default DiGraph)
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Graph type to create. If graph instance, then cleared before populated.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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References
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----------
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.. [1] P. L. Krapivsky and S. Redner,
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Network Growth by Copying,
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Phys. Rev. E, 71, 036118, 2005k.},
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"""
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G = empty_graph(1, create_using, default=nx.DiGraph)
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if not G.is_directed():
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raise nx.NetworkXError("create_using must indicate a Directed Graph")
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if n == 1:
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return G
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for source in range(1, n):
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target = seed.randrange(0, source)
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for succ in G.successors(target):
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G.add_edge(source, succ)
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G.add_edge(source, target)
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return G
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@py_random_state(7)
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def scale_free_graph(
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n,
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alpha=0.41,
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beta=0.54,
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gamma=0.05,
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delta_in=0.2,
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delta_out=0,
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create_using=None,
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seed=None,
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):
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"""Returns a scale-free directed graph.
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Parameters
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----------
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n : integer
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Number of nodes in graph
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alpha : float
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Probability for adding a new node connected to an existing node
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chosen randomly according to the in-degree distribution.
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beta : float
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Probability for adding an edge between two existing nodes.
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One existing node is chosen randomly according the in-degree
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distribution and the other chosen randomly according to the out-degree
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distribution.
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gamma : float
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Probability for adding a new node connected to an existing node
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chosen randomly according to the out-degree distribution.
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delta_in : float
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Bias for choosing nodes from in-degree distribution.
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delta_out : float
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Bias for choosing nodes from out-degree distribution.
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create_using : NetworkX graph constructor, optional
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The default is a MultiDiGraph 3-cycle.
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If a graph instance, use it without clearing first.
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If a graph constructor, call it to construct an empty graph.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Examples
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--------
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Create a scale-free graph on one hundred nodes::
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>>> G = nx.scale_free_graph(100)
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Notes
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-----
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The sum of `alpha`, `beta`, and `gamma` must be 1.
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References
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----------
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.. [1] B. Bollobás, C. Borgs, J. Chayes, and O. Riordan,
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Directed scale-free graphs,
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Proceedings of the fourteenth annual ACM-SIAM Symposium on
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Discrete Algorithms, 132--139, 2003.
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"""
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def _choose_node(candidates, node_list, delta):
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if delta > 0:
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bias_sum = len(node_list) * delta
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p_delta = bias_sum / (bias_sum + len(candidates))
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if seed.random() < p_delta:
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return seed.choice(node_list)
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return seed.choice(candidates)
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if create_using is None or not hasattr(create_using, "_adj"):
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# start with 3-cycle
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G = nx.empty_graph(3, create_using, default=nx.MultiDiGraph)
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G.add_edges_from([(0, 1), (1, 2), (2, 0)])
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else:
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G = create_using
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if not (G.is_directed() and G.is_multigraph()):
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raise nx.NetworkXError("MultiDiGraph required in create_using")
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if alpha <= 0:
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raise ValueError("alpha must be > 0.")
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if beta <= 0:
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raise ValueError("beta must be > 0.")
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if gamma <= 0:
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raise ValueError("gamma must be > 0.")
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if abs(alpha + beta + gamma - 1.0) >= 1e-9:
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raise ValueError("alpha+beta+gamma must equal 1.")
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if delta_in < 0:
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raise ValueError("delta_in must be >= 0.")
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if delta_out < 0:
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raise ValueError("delta_out must be >= 0.")
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# pre-populate degree states
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vs = sum((count * [idx] for idx, count in G.out_degree()), [])
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ws = sum((count * [idx] for idx, count in G.in_degree()), [])
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# pre-populate node state
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node_list = list(G.nodes())
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# see if there already are number-based nodes
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numeric_nodes = [n for n in node_list if isinstance(n, numbers.Number)]
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if len(numeric_nodes) > 0:
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# set cursor for new nodes appropriately
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cursor = max(int(n.real) for n in numeric_nodes) + 1
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else:
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# or start at zero
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cursor = 0
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while len(G) < n:
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r = seed.random()
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# random choice in alpha,beta,gamma ranges
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if r < alpha:
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# alpha
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# add new node v
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v = cursor
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cursor += 1
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# also add to node state
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node_list.append(v)
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# choose w according to in-degree and delta_in
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w = _choose_node(ws, node_list, delta_in)
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elif r < alpha + beta:
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# beta
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# choose v according to out-degree and delta_out
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v = _choose_node(vs, node_list, delta_out)
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# choose w according to in-degree and delta_in
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w = _choose_node(ws, node_list, delta_in)
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else:
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# gamma
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# choose v according to out-degree and delta_out
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v = _choose_node(vs, node_list, delta_out)
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# add new node w
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w = cursor
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cursor += 1
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# also add to node state
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node_list.append(w)
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# add edge to graph
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G.add_edge(v, w)
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# update degree states
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vs.append(v)
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ws.append(w)
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return G
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@py_random_state(4)
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def random_uniform_k_out_graph(n, k, self_loops=True, with_replacement=True, seed=None):
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"""Returns a random `k`-out graph with uniform attachment.
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A random `k`-out graph with uniform attachment is a multidigraph
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generated by the following algorithm. For each node *u*, choose
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`k` nodes *v* uniformly at random (with replacement). Add a
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directed edge joining *u* to *v*.
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Parameters
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----------
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n : int
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The number of nodes in the returned graph.
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k : int
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The out-degree of each node in the returned graph.
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self_loops : bool
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If True, self-loops are allowed when generating the graph.
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with_replacement : bool
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If True, neighbors are chosen with replacement and the
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returned graph will be a directed multigraph. Otherwise,
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neighbors are chosen without replacement and the returned graph
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will be a directed graph.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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NetworkX graph
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A `k`-out-regular directed graph generated according to the
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above algorithm. It will be a multigraph if and only if
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`with_replacement` is True.
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Raises
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------
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ValueError
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If `with_replacement` is False and `k` is greater than
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`n`.
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See also
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--------
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random_k_out_graph
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Notes
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-----
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The return digraph or multidigraph may not be strongly connected, or
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even weakly connected.
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If `with_replacement` is True, this function is similar to
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:func:`random_k_out_graph`, if that function had parameter `alpha`
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set to positive infinity.
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"""
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if with_replacement:
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create_using = nx.MultiDiGraph()
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def sample(v, nodes):
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if not self_loops:
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nodes = nodes - {v}
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return (seed.choice(list(nodes)) for i in range(k))
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else:
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create_using = nx.DiGraph()
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def sample(v, nodes):
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if not self_loops:
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nodes = nodes - {v}
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return seed.sample(list(nodes), k)
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G = nx.empty_graph(n, create_using)
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nodes = set(G)
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for u in G:
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G.add_edges_from((u, v) for v in sample(u, nodes))
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return G
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@py_random_state(4)
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def random_k_out_graph(n, k, alpha, self_loops=True, seed=None):
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"""Returns a random `k`-out graph with preferential attachment.
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A random `k`-out graph with preferential attachment is a
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multidigraph generated by the following algorithm.
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1. Begin with an empty digraph, and initially set each node to have
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weight `alpha`.
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2. Choose a node `u` with out-degree less than `k` uniformly at
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random.
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3. Choose a node `v` from with probability proportional to its
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weight.
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4. Add a directed edge from `u` to `v`, and increase the weight
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of `v` by one.
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5. If each node has out-degree `k`, halt, otherwise repeat from
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step 2.
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For more information on this model of random graph, see [1].
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Parameters
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----------
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n : int
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The number of nodes in the returned graph.
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k : int
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The out-degree of each node in the returned graph.
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alpha : float
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A positive :class:`float` representing the initial weight of
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each vertex. A higher number means that in step 3 above, nodes
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will be chosen more like a true uniformly random sample, and a
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lower number means that nodes are more likely to be chosen as
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their in-degree increases. If this parameter is not positive, a
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:exc:`ValueError` is raised.
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self_loops : bool
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If True, self-loops are allowed when generating the graph.
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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:class:`~networkx.classes.MultiDiGraph`
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A `k`-out-regular multidigraph generated according to the above
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algorithm.
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Raises
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------
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ValueError
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If `alpha` is not positive.
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Notes
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-----
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The returned multidigraph may not be strongly connected, or even
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weakly connected.
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References
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----------
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[1]: Peterson, Nicholas R., and Boris Pittel.
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"Distance between two random `k`-out digraphs, with and without
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preferential attachment."
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arXiv preprint arXiv:1311.5961 (2013).
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<https://arxiv.org/abs/1311.5961>
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"""
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if alpha < 0:
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raise ValueError("alpha must be positive")
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G = nx.empty_graph(n, create_using=nx.MultiDiGraph)
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weights = Counter({v: alpha for v in G})
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for i in range(k * n):
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u = seed.choice([v for v, d in G.out_degree() if d < k])
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# If self-loops are not allowed, make the source node `u` have
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# weight zero.
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if not self_loops:
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adjustment = Counter({u: weights[u]})
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else:
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adjustment = Counter()
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v = weighted_choice(weights - adjustment, seed=seed)
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G.add_edge(u, v)
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weights[v] += 1
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return G
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