1295 lines
42 KiB
Python
1295 lines
42 KiB
Python
"""
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Generators for random graphs.
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"""
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import itertools
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import math
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import networkx as nx
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from networkx.utils import py_random_state
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from .classic import empty_graph, path_graph, complete_graph
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from .degree_seq import degree_sequence_tree
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from collections import defaultdict
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__all__ = [
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"fast_gnp_random_graph",
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"gnp_random_graph",
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"dense_gnm_random_graph",
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"gnm_random_graph",
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"erdos_renyi_graph",
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"binomial_graph",
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"newman_watts_strogatz_graph",
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"watts_strogatz_graph",
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"connected_watts_strogatz_graph",
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"random_regular_graph",
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"barabasi_albert_graph",
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"dual_barabasi_albert_graph",
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"extended_barabasi_albert_graph",
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"powerlaw_cluster_graph",
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"random_lobster",
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"random_shell_graph",
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"random_powerlaw_tree",
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"random_powerlaw_tree_sequence",
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"random_kernel_graph",
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]
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@py_random_state(2)
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def fast_gnp_random_graph(n, p, seed=None, directed=False):
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"""Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph or
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a binomial graph.
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Parameters
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----------
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n : int
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The number of nodes.
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p : float
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Probability for edge creation.
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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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directed : bool, optional (default=False)
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If True, this function returns a directed graph.
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Notes
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-----
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The $G_{n,p}$ graph algorithm chooses each of the $[n (n - 1)] / 2$
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(undirected) or $n (n - 1)$ (directed) possible edges with probability $p$.
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This algorithm [1]_ runs in $O(n + m)$ time, where `m` is the expected number of
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edges, which equals $p n (n - 1) / 2$. This should be faster than
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:func:`gnp_random_graph` when $p$ is small and the expected number of edges
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is small (that is, the graph is sparse).
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See Also
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--------
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gnp_random_graph
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References
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----------
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.. [1] Vladimir Batagelj and Ulrik Brandes,
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"Efficient generation of large random networks",
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Phys. Rev. E, 71, 036113, 2005.
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"""
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G = empty_graph(n)
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if p <= 0 or p >= 1:
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return nx.gnp_random_graph(n, p, seed=seed, directed=directed)
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w = -1
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lp = math.log(1.0 - p)
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if directed:
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G = nx.DiGraph(G)
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# Nodes in graph are from 0,n-1 (start with v as the first node index).
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v = 0
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while v < n:
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lr = math.log(1.0 - seed.random())
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w = w + 1 + int(lr / lp)
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if v == w: # avoid self loops
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w = w + 1
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while v < n <= w:
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w = w - n
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v = v + 1
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if v == w: # avoid self loops
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w = w + 1
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if v < n:
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G.add_edge(v, w)
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else:
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# Nodes in graph are from 0,n-1 (start with v as the second node index).
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v = 1
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while v < n:
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lr = math.log(1.0 - seed.random())
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w = w + 1 + int(lr / lp)
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while w >= v and v < n:
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w = w - v
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v = v + 1
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if v < n:
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G.add_edge(v, w)
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return G
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@py_random_state(2)
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def gnp_random_graph(n, p, seed=None, directed=False):
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"""Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph
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or a binomial graph.
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The $G_{n,p}$ model chooses each of the possible edges with probability $p$.
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Parameters
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----------
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n : int
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The number of nodes.
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p : float
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Probability for edge creation.
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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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directed : bool, optional (default=False)
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If True, this function returns a directed graph.
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See Also
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--------
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fast_gnp_random_graph
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Notes
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-----
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This algorithm [2]_ runs in $O(n^2)$ time. For sparse graphs (that is, for
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small values of $p$), :func:`fast_gnp_random_graph` is a faster algorithm.
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:func:`binomial_graph` and :func:`erdos_renyi_graph` are
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aliases for :func:`gnp_random_graph`.
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>>> nx.binomial_graph is nx.gnp_random_graph
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True
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>>> nx.erdos_renyi_graph is nx.gnp_random_graph
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True
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References
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----------
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.. [1] P. Erdős and A. Rényi, On Random Graphs, Publ. Math. 6, 290 (1959).
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.. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
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"""
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if directed:
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edges = itertools.permutations(range(n), 2)
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G = nx.DiGraph()
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else:
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edges = itertools.combinations(range(n), 2)
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G = nx.Graph()
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G.add_nodes_from(range(n))
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if p <= 0:
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return G
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if p >= 1:
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return complete_graph(n, create_using=G)
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for e in edges:
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if seed.random() < p:
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G.add_edge(*e)
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return G
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# add some aliases to common names
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binomial_graph = gnp_random_graph
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erdos_renyi_graph = gnp_random_graph
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@py_random_state(2)
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def dense_gnm_random_graph(n, m, seed=None):
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"""Returns a $G_{n,m}$ random graph.
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In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
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of all graphs with $n$ nodes and $m$ edges.
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This algorithm should be faster than :func:`gnm_random_graph` for dense
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graphs.
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Parameters
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----------
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n : int
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The number of nodes.
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m : int
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The number of edges.
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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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See Also
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--------
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gnm_random_graph()
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Notes
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-----
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Algorithm by Keith M. Briggs Mar 31, 2006.
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Inspired by Knuth's Algorithm S (Selection sampling technique),
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in section 3.4.2 of [1]_.
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References
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----------
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.. [1] Donald E. Knuth, The Art of Computer Programming,
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Volume 2/Seminumerical algorithms, Third Edition, Addison-Wesley, 1997.
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"""
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mmax = n * (n - 1) / 2
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if m >= mmax:
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G = complete_graph(n)
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else:
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G = empty_graph(n)
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if n == 1 or m >= mmax:
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return G
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u = 0
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v = 1
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t = 0
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k = 0
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while True:
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if seed.randrange(mmax - t) < m - k:
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G.add_edge(u, v)
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k += 1
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if k == m:
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return G
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t += 1
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v += 1
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if v == n: # go to next row of adjacency matrix
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u += 1
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v = u + 1
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@py_random_state(2)
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def gnm_random_graph(n, m, seed=None, directed=False):
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"""Returns a $G_{n,m}$ random graph.
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In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
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of all graphs with $n$ nodes and $m$ edges.
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This algorithm should be faster than :func:`dense_gnm_random_graph` for
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sparse graphs.
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Parameters
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----------
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n : int
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The number of nodes.
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m : int
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The number of edges.
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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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directed : bool, optional (default=False)
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If True return a directed graph
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See also
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--------
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dense_gnm_random_graph
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"""
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if directed:
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G = nx.DiGraph()
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else:
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G = nx.Graph()
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G.add_nodes_from(range(n))
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if n == 1:
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return G
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max_edges = n * (n - 1)
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if not directed:
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max_edges /= 2.0
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if m >= max_edges:
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return complete_graph(n, create_using=G)
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nlist = list(G)
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edge_count = 0
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while edge_count < m:
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# generate random edge,u,v
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u = seed.choice(nlist)
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v = seed.choice(nlist)
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if u == v or G.has_edge(u, v):
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continue
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else:
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G.add_edge(u, v)
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edge_count = edge_count + 1
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return G
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@py_random_state(3)
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def newman_watts_strogatz_graph(n, k, p, seed=None):
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"""Returns a Newman–Watts–Strogatz small-world graph.
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Parameters
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----------
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n : int
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The number of nodes.
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k : int
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Each node is joined with its `k` nearest neighbors in a ring
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topology.
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p : float
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The probability of adding a new edge for each edge.
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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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Notes
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-----
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First create a ring over $n$ nodes [1]_. Then each node in the ring is
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connected with its $k$ nearest neighbors (or $k - 1$ neighbors if $k$
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is odd). Then shortcuts are created by adding new edges as follows: for
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each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest
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neighbors" with probability $p$ add a new edge $(u, w)$ with
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randomly-chosen existing node $w$. In contrast with
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:func:`watts_strogatz_graph`, no edges are removed.
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See Also
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--------
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watts_strogatz_graph()
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References
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----------
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.. [1] M. E. J. Newman and D. J. Watts,
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Renormalization group analysis of the small-world network model,
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Physics Letters A, 263, 341, 1999.
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||
https://doi.org/10.1016/S0375-9601(99)00757-4
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"""
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if k > n:
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raise nx.NetworkXError("k>=n, choose smaller k or larger n")
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# If k == n the graph return is a complete graph
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if k == n:
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return nx.complete_graph(n)
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G = empty_graph(n)
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nlist = list(G.nodes())
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fromv = nlist
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# connect the k/2 neighbors
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for j in range(1, k // 2 + 1):
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tov = fromv[j:] + fromv[0:j] # the first j are now last
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for i in range(len(fromv)):
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G.add_edge(fromv[i], tov[i])
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# for each edge u-v, with probability p, randomly select existing
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# node w and add new edge u-w
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e = list(G.edges())
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for (u, v) in e:
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if seed.random() < p:
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w = seed.choice(nlist)
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# no self-loops and reject if edge u-w exists
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# is that the correct NWS model?
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while w == u or G.has_edge(u, w):
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w = seed.choice(nlist)
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if G.degree(u) >= n - 1:
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break # skip this rewiring
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else:
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G.add_edge(u, w)
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return G
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@py_random_state(3)
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def watts_strogatz_graph(n, k, p, seed=None):
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"""Returns a Watts–Strogatz small-world graph.
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||
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Parameters
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----------
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n : int
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The number of nodes
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k : int
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Each node is joined with its `k` nearest neighbors in a ring
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topology.
|
||
p : float
|
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The probability of rewiring each edge
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||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
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See Also
|
||
--------
|
||
newman_watts_strogatz_graph()
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connected_watts_strogatz_graph()
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||
|
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Notes
|
||
-----
|
||
First create a ring over $n$ nodes [1]_. Then each node in the ring is joined
|
||
to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd).
|
||
Then shortcuts are created by replacing some edges as follows: for each
|
||
edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors"
|
||
with probability $p$ replace it with a new edge $(u, w)$ with uniformly
|
||
random choice of existing node $w$.
|
||
|
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In contrast with :func:`newman_watts_strogatz_graph`, the random rewiring
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does not increase the number of edges. The rewired graph is not guaranteed
|
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to be connected as in :func:`connected_watts_strogatz_graph`.
|
||
|
||
References
|
||
----------
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.. [1] Duncan J. Watts and Steven H. Strogatz,
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||
Collective dynamics of small-world networks,
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Nature, 393, pp. 440--442, 1998.
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||
"""
|
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if k > n:
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raise nx.NetworkXError("k>n, choose smaller k or larger n")
|
||
|
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# If k == n, the graph is complete not Watts-Strogatz
|
||
if k == n:
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return nx.complete_graph(n)
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||
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G = nx.Graph()
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nodes = list(range(n)) # nodes are labeled 0 to n-1
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# connect each node to k/2 neighbors
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for j in range(1, k // 2 + 1):
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targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list
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G.add_edges_from(zip(nodes, targets))
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# rewire edges from each node
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# loop over all nodes in order (label) and neighbors in order (distance)
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# no self loops or multiple edges allowed
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for j in range(1, k // 2 + 1): # outer loop is neighbors
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targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list
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# inner loop in node order
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||
for u, v in zip(nodes, targets):
|
||
if seed.random() < p:
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||
w = seed.choice(nodes)
|
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# Enforce no self-loops or multiple edges
|
||
while w == u or G.has_edge(u, w):
|
||
w = seed.choice(nodes)
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if G.degree(u) >= n - 1:
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break # skip this rewiring
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||
else:
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||
G.remove_edge(u, v)
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G.add_edge(u, w)
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return G
|
||
|
||
|
||
@py_random_state(4)
|
||
def connected_watts_strogatz_graph(n, k, p, tries=100, seed=None):
|
||
"""Returns a connected Watts–Strogatz small-world graph.
|
||
|
||
Attempts to generate a connected graph by repeated generation of
|
||
Watts–Strogatz small-world graphs. An exception is raised if the maximum
|
||
number of tries is exceeded.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
The number of nodes
|
||
k : int
|
||
Each node is joined with its `k` nearest neighbors in a ring
|
||
topology.
|
||
p : float
|
||
The probability of rewiring each edge
|
||
tries : int
|
||
Number of attempts to generate a connected graph.
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Notes
|
||
-----
|
||
First create a ring over $n$ nodes [1]_. Then each node in the ring is joined
|
||
to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd).
|
||
Then shortcuts are created by replacing some edges as follows: for each
|
||
edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors"
|
||
with probability $p$ replace it with a new edge $(u, w)$ with uniformly
|
||
random choice of existing node $w$.
|
||
The entire process is repeated until a connected graph results.
|
||
|
||
See Also
|
||
--------
|
||
newman_watts_strogatz_graph()
|
||
watts_strogatz_graph()
|
||
|
||
References
|
||
----------
|
||
.. [1] Duncan J. Watts and Steven H. Strogatz,
|
||
Collective dynamics of small-world networks,
|
||
Nature, 393, pp. 440--442, 1998.
|
||
"""
|
||
for i in range(tries):
|
||
# seed is an RNG so should change sequence each call
|
||
G = watts_strogatz_graph(n, k, p, seed)
|
||
if nx.is_connected(G):
|
||
return G
|
||
raise nx.NetworkXError("Maximum number of tries exceeded")
|
||
|
||
|
||
@py_random_state(2)
|
||
def random_regular_graph(d, n, seed=None):
|
||
r"""Returns a random $d$-regular graph on $n$ nodes.
|
||
|
||
The resulting graph has no self-loops or parallel edges.
|
||
|
||
Parameters
|
||
----------
|
||
d : int
|
||
The degree of each node.
|
||
n : integer
|
||
The number of nodes. The value of $n \times d$ must be even.
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Notes
|
||
-----
|
||
The nodes are numbered from $0$ to $n - 1$.
|
||
|
||
Kim and Vu's paper [2]_ shows that this algorithm samples in an
|
||
asymptotically uniform way from the space of random graphs when
|
||
$d = O(n^{1 / 3 - \epsilon})$.
|
||
|
||
Raises
|
||
------
|
||
|
||
NetworkXError
|
||
If $n \times d$ is odd or $d$ is greater than or equal to $n$.
|
||
|
||
References
|
||
----------
|
||
.. [1] A. Steger and N. Wormald,
|
||
Generating random regular graphs quickly,
|
||
Probability and Computing 8 (1999), 377-396, 1999.
|
||
http://citeseer.ist.psu.edu/steger99generating.html
|
||
|
||
.. [2] Jeong Han Kim and Van H. Vu,
|
||
Generating random regular graphs,
|
||
Proceedings of the thirty-fifth ACM symposium on Theory of computing,
|
||
San Diego, CA, USA, pp 213--222, 2003.
|
||
http://portal.acm.org/citation.cfm?id=780542.780576
|
||
"""
|
||
if (n * d) % 2 != 0:
|
||
raise nx.NetworkXError("n * d must be even")
|
||
|
||
if not 0 <= d < n:
|
||
raise nx.NetworkXError("the 0 <= d < n inequality must be satisfied")
|
||
|
||
if d == 0:
|
||
return empty_graph(n)
|
||
|
||
def _suitable(edges, potential_edges):
|
||
# Helper subroutine to check if there are suitable edges remaining
|
||
# If False, the generation of the graph has failed
|
||
if not potential_edges:
|
||
return True
|
||
for s1 in potential_edges:
|
||
for s2 in potential_edges:
|
||
# Two iterators on the same dictionary are guaranteed
|
||
# to visit it in the same order if there are no
|
||
# intervening modifications.
|
||
if s1 == s2:
|
||
# Only need to consider s1-s2 pair one time
|
||
break
|
||
if s1 > s2:
|
||
s1, s2 = s2, s1
|
||
if (s1, s2) not in edges:
|
||
return True
|
||
return False
|
||
|
||
def _try_creation():
|
||
# Attempt to create an edge set
|
||
|
||
edges = set()
|
||
stubs = list(range(n)) * d
|
||
|
||
while stubs:
|
||
potential_edges = defaultdict(lambda: 0)
|
||
seed.shuffle(stubs)
|
||
stubiter = iter(stubs)
|
||
for s1, s2 in zip(stubiter, stubiter):
|
||
if s1 > s2:
|
||
s1, s2 = s2, s1
|
||
if s1 != s2 and ((s1, s2) not in edges):
|
||
edges.add((s1, s2))
|
||
else:
|
||
potential_edges[s1] += 1
|
||
potential_edges[s2] += 1
|
||
|
||
if not _suitable(edges, potential_edges):
|
||
return None # failed to find suitable edge set
|
||
|
||
stubs = [
|
||
node
|
||
for node, potential in potential_edges.items()
|
||
for _ in range(potential)
|
||
]
|
||
return edges
|
||
|
||
# Even though a suitable edge set exists,
|
||
# the generation of such a set is not guaranteed.
|
||
# Try repeatedly to find one.
|
||
edges = _try_creation()
|
||
while edges is None:
|
||
edges = _try_creation()
|
||
|
||
G = nx.Graph()
|
||
G.add_edges_from(edges)
|
||
|
||
return G
|
||
|
||
|
||
def _random_subset(seq, m, rng):
|
||
""" Return m unique elements from seq.
|
||
|
||
This differs from random.sample which can return repeated
|
||
elements if seq holds repeated elements.
|
||
|
||
Note: rng is a random.Random or numpy.random.RandomState instance.
|
||
"""
|
||
targets = set()
|
||
while len(targets) < m:
|
||
x = rng.choice(seq)
|
||
targets.add(x)
|
||
return targets
|
||
|
||
|
||
@py_random_state(2)
|
||
def barabasi_albert_graph(n, m, seed=None):
|
||
"""Returns a random graph according to the Barabási–Albert preferential
|
||
attachment model.
|
||
|
||
A graph of $n$ nodes is grown by attaching new nodes each with $m$
|
||
edges that are preferentially attached to existing nodes with high degree.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
Number of nodes
|
||
m : int
|
||
Number of edges to attach from a new node to existing nodes
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Returns
|
||
-------
|
||
G : Graph
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If `m` does not satisfy ``1 <= m < n``.
|
||
|
||
References
|
||
----------
|
||
.. [1] A. L. Barabási and R. Albert "Emergence of scaling in
|
||
random networks", Science 286, pp 509-512, 1999.
|
||
"""
|
||
|
||
if m < 1 or m >= n:
|
||
raise nx.NetworkXError(
|
||
f"Barabási–Albert network must have m >= 1 and m < n, m = {m}, n = {n}"
|
||
)
|
||
|
||
# Add m initial nodes (m0 in barabasi-speak)
|
||
G = empty_graph(m)
|
||
# Target nodes for new edges
|
||
targets = list(range(m))
|
||
# List of existing nodes, with nodes repeated once for each adjacent edge
|
||
repeated_nodes = []
|
||
# Start adding the other n-m nodes. The first node is m.
|
||
source = m
|
||
while source < n:
|
||
# Add edges to m nodes from the source.
|
||
G.add_edges_from(zip([source] * m, targets))
|
||
# Add one node to the list for each new edge just created.
|
||
repeated_nodes.extend(targets)
|
||
# And the new node "source" has m edges to add to the list.
|
||
repeated_nodes.extend([source] * m)
|
||
# Now choose m unique nodes from the existing nodes
|
||
# Pick uniformly from repeated_nodes (preferential attachment)
|
||
targets = _random_subset(repeated_nodes, m, seed)
|
||
source += 1
|
||
return G
|
||
|
||
|
||
@py_random_state(4)
|
||
def dual_barabasi_albert_graph(n, m1, m2, p, seed=None):
|
||
"""Returns a random graph according to the dual Barabási–Albert preferential
|
||
attachment model.
|
||
|
||
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
|
||
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
|
||
are preferentially attached to existing nodes with high degree.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
Number of nodes
|
||
m1 : int
|
||
Number of edges to attach from a new node to existing nodes with probability $p$
|
||
m2 : int
|
||
Number of edges to attach from a new node to existing nodes with probability $1-p$
|
||
p : float
|
||
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Returns
|
||
-------
|
||
G : Graph
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n`` or `p` does not satisfy ``0 <= p <= 1``.
|
||
|
||
References
|
||
----------
|
||
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
|
||
"""
|
||
|
||
if m1 < 1 or m1 >= n:
|
||
raise nx.NetworkXError(
|
||
f"Dual Barabási–Albert network must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
|
||
)
|
||
if m2 < 1 or m2 >= n:
|
||
raise nx.NetworkXError(
|
||
f"Dual Barabási–Albert network must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
|
||
)
|
||
if p < 0 or p > 1:
|
||
raise nx.NetworkXError(
|
||
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
|
||
)
|
||
|
||
# For simplicity, if p == 0 or 1, just return BA
|
||
if p == 1:
|
||
return barabasi_albert_graph(n, m1, seed)
|
||
elif p == 0:
|
||
return barabasi_albert_graph(n, m2, seed)
|
||
|
||
# Add max(m1,m2) initial nodes (m0 in barabasi-speak)
|
||
G = empty_graph(max(m1, m2))
|
||
# Target nodes for new edges
|
||
targets = list(range(max(m1, m2)))
|
||
# List of existing nodes, with nodes repeated once for each adjacent edge
|
||
repeated_nodes = []
|
||
# Start adding the remaining nodes.
|
||
source = max(m1, m2)
|
||
# Pick which m to use first time (m1 or m2)
|
||
if seed.random() < p:
|
||
m = m1
|
||
else:
|
||
m = m2
|
||
while source < n:
|
||
# Add edges to m nodes from the source.
|
||
G.add_edges_from(zip([source] * m, targets))
|
||
# Add one node to the list for each new edge just created.
|
||
repeated_nodes.extend(targets)
|
||
# And the new node "source" has m edges to add to the list.
|
||
repeated_nodes.extend([source] * m)
|
||
# Pick which m to use next time (m1 or m2)
|
||
if seed.random() < p:
|
||
m = m1
|
||
else:
|
||
m = m2
|
||
# Now choose m unique nodes from the existing nodes
|
||
# Pick uniformly from repeated_nodes (preferential attachment)
|
||
targets = _random_subset(repeated_nodes, m, seed)
|
||
source += 1
|
||
return G
|
||
|
||
|
||
@py_random_state(4)
|
||
def extended_barabasi_albert_graph(n, m, p, q, seed=None):
|
||
"""Returns an extended Barabási–Albert model graph.
|
||
|
||
An extended Barabási–Albert model graph is a random graph constructed
|
||
using preferential attachment. The extended model allows new edges,
|
||
rewired edges or new nodes. Based on the probabilities $p$ and $q$
|
||
with $p + q < 1$, the growing behavior of the graph is determined as:
|
||
|
||
1) With $p$ probability, $m$ new edges are added to the graph,
|
||
starting from randomly chosen existing nodes and attached preferentially at the other end.
|
||
|
||
2) With $q$ probability, $m$ existing edges are rewired
|
||
by randomly choosing an edge and rewiring one end to a preferentially chosen node.
|
||
|
||
3) With $(1 - p - q)$ probability, $m$ new nodes are added to the graph
|
||
with edges attached preferentially.
|
||
|
||
When $p = q = 0$, the model behaves just like the Barabási–Alber model.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
Number of nodes
|
||
m : int
|
||
Number of edges with which a new node attaches to existing nodes
|
||
p : float
|
||
Probability value for adding an edge between existing nodes. p + q < 1
|
||
q : float
|
||
Probability value of rewiring of existing edges. p + q < 1
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Returns
|
||
-------
|
||
G : Graph
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If `m` does not satisfy ``1 <= m < n`` or ``1 >= p + q``
|
||
|
||
References
|
||
----------
|
||
.. [1] Albert, R., & Barabási, A. L. (2000)
|
||
Topology of evolving networks: local events and universality
|
||
Physical review letters, 85(24), 5234.
|
||
"""
|
||
if m < 1 or m >= n:
|
||
msg = f"Extended Barabasi-Albert network needs m>=1 and m<n, m={m}, n={n}"
|
||
raise nx.NetworkXError(msg)
|
||
if p + q >= 1:
|
||
msg = f"Extended Barabasi-Albert network needs p + q <= 1, p={p}, q={q}"
|
||
raise nx.NetworkXError(msg)
|
||
|
||
# Add m initial nodes (m0 in barabasi-speak)
|
||
G = empty_graph(m)
|
||
|
||
# List of nodes to represent the preferential attachment random selection.
|
||
# At the creation of the graph, all nodes are added to the list
|
||
# so that even nodes that are not connected have a chance to get selected,
|
||
# for rewiring and adding of edges.
|
||
# With each new edge, nodes at the ends of the edge are added to the list.
|
||
attachment_preference = []
|
||
attachment_preference.extend(range(m))
|
||
|
||
# Start adding the other n-m nodes. The first node is m.
|
||
new_node = m
|
||
while new_node < n:
|
||
a_probability = seed.random()
|
||
|
||
# Total number of edges of a Clique of all the nodes
|
||
clique_degree = len(G) - 1
|
||
clique_size = (len(G) * clique_degree) / 2
|
||
|
||
# Adding m new edges, if there is room to add them
|
||
if a_probability < p and G.size() <= clique_size - m:
|
||
# Select the nodes where an edge can be added
|
||
elligible_nodes = [nd for nd, deg in G.degree() if deg < clique_degree]
|
||
for i in range(m):
|
||
# Choosing a random source node from elligible_nodes
|
||
src_node = seed.choice(elligible_nodes)
|
||
|
||
# Picking a possible node that is not 'src_node' or
|
||
# neighbor with 'src_node', with preferential attachment
|
||
prohibited_nodes = list(G[src_node])
|
||
prohibited_nodes.append(src_node)
|
||
# This will raise an exception if the sequence is empty
|
||
dest_node = seed.choice(
|
||
[nd for nd in attachment_preference if nd not in prohibited_nodes]
|
||
)
|
||
# Adding the new edge
|
||
G.add_edge(src_node, dest_node)
|
||
|
||
# Appending both nodes to add to their preferential attachment
|
||
attachment_preference.append(src_node)
|
||
attachment_preference.append(dest_node)
|
||
|
||
# Adjusting the elligible nodes. Degree may be saturated.
|
||
if G.degree(src_node) == clique_degree:
|
||
elligible_nodes.remove(src_node)
|
||
if (
|
||
G.degree(dest_node) == clique_degree
|
||
and dest_node in elligible_nodes
|
||
):
|
||
elligible_nodes.remove(dest_node)
|
||
|
||
# Rewiring m edges, if there are enough edges
|
||
elif p <= a_probability < (p + q) and m <= G.size() < clique_size:
|
||
# Selecting nodes that have at least 1 edge but that are not
|
||
# fully connected to ALL other nodes (center of star).
|
||
# These nodes are the pivot nodes of the edges to rewire
|
||
elligible_nodes = [nd for nd, deg in G.degree() if 0 < deg < clique_degree]
|
||
for i in range(m):
|
||
# Choosing a random source node
|
||
node = seed.choice(elligible_nodes)
|
||
|
||
# The available nodes do have a neighbor at least.
|
||
neighbor_nodes = list(G[node])
|
||
|
||
# Choosing the other end that will get dettached
|
||
src_node = seed.choice(neighbor_nodes)
|
||
|
||
# Picking a target node that is not 'node' or
|
||
# neighbor with 'node', with preferential attachment
|
||
neighbor_nodes.append(node)
|
||
dest_node = seed.choice(
|
||
[nd for nd in attachment_preference if nd not in neighbor_nodes]
|
||
)
|
||
# Rewire
|
||
G.remove_edge(node, src_node)
|
||
G.add_edge(node, dest_node)
|
||
|
||
# Adjusting the preferential attachment list
|
||
attachment_preference.remove(src_node)
|
||
attachment_preference.append(dest_node)
|
||
|
||
# Adjusting the elligible nodes.
|
||
# nodes may be saturated or isolated.
|
||
if G.degree(src_node) == 0 and src_node in elligible_nodes:
|
||
elligible_nodes.remove(src_node)
|
||
if dest_node in elligible_nodes:
|
||
if G.degree(dest_node) == clique_degree:
|
||
elligible_nodes.remove(dest_node)
|
||
else:
|
||
if G.degree(dest_node) == 1:
|
||
elligible_nodes.append(dest_node)
|
||
|
||
# Adding new node with m edges
|
||
else:
|
||
# Select the edges' nodes by preferential attachment
|
||
targets = _random_subset(attachment_preference, m, seed)
|
||
G.add_edges_from(zip([new_node] * m, targets))
|
||
|
||
# Add one node to the list for each new edge just created.
|
||
attachment_preference.extend(targets)
|
||
# The new node has m edges to it, plus itself: m + 1
|
||
attachment_preference.extend([new_node] * (m + 1))
|
||
new_node += 1
|
||
return G
|
||
|
||
|
||
@py_random_state(3)
|
||
def powerlaw_cluster_graph(n, m, p, seed=None):
|
||
"""Holme and Kim algorithm for growing graphs with powerlaw
|
||
degree distribution and approximate average clustering.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
the number of nodes
|
||
m : int
|
||
the number of random edges to add for each new node
|
||
p : float,
|
||
Probability of adding a triangle after adding a random edge
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Notes
|
||
-----
|
||
The average clustering has a hard time getting above a certain
|
||
cutoff that depends on `m`. This cutoff is often quite low. The
|
||
transitivity (fraction of triangles to possible triangles) seems to
|
||
decrease with network size.
|
||
|
||
It is essentially the Barabási–Albert (BA) growth model with an
|
||
extra step that each random edge is followed by a chance of
|
||
making an edge to one of its neighbors too (and thus a triangle).
|
||
|
||
This algorithm improves on BA in the sense that it enables a
|
||
higher average clustering to be attained if desired.
|
||
|
||
It seems possible to have a disconnected graph with this algorithm
|
||
since the initial `m` nodes may not be all linked to a new node
|
||
on the first iteration like the BA model.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If `m` does not satisfy ``1 <= m <= n`` or `p` does not
|
||
satisfy ``0 <= p <= 1``.
|
||
|
||
References
|
||
----------
|
||
.. [1] P. Holme and B. J. Kim,
|
||
"Growing scale-free networks with tunable clustering",
|
||
Phys. Rev. E, 65, 026107, 2002.
|
||
"""
|
||
|
||
if m < 1 or n < m:
|
||
raise nx.NetworkXError(f"NetworkXError must have m>1 and m<n, m={m},n={n}")
|
||
|
||
if p > 1 or p < 0:
|
||
raise nx.NetworkXError(f"NetworkXError p must be in [0,1], p={p}")
|
||
|
||
G = empty_graph(m) # add m initial nodes (m0 in barabasi-speak)
|
||
repeated_nodes = list(G.nodes()) # list of existing nodes to sample from
|
||
# with nodes repeated once for each adjacent edge
|
||
source = m # next node is m
|
||
while source < n: # Now add the other n-1 nodes
|
||
possible_targets = _random_subset(repeated_nodes, m, seed)
|
||
# do one preferential attachment for new node
|
||
target = possible_targets.pop()
|
||
G.add_edge(source, target)
|
||
repeated_nodes.append(target) # add one node to list for each new link
|
||
count = 1
|
||
while count < m: # add m-1 more new links
|
||
if seed.random() < p: # clustering step: add triangle
|
||
neighborhood = [
|
||
nbr
|
||
for nbr in G.neighbors(target)
|
||
if not G.has_edge(source, nbr) and not nbr == source
|
||
]
|
||
if neighborhood: # if there is a neighbor without a link
|
||
nbr = seed.choice(neighborhood)
|
||
G.add_edge(source, nbr) # add triangle
|
||
repeated_nodes.append(nbr)
|
||
count = count + 1
|
||
continue # go to top of while loop
|
||
# else do preferential attachment step if above fails
|
||
target = possible_targets.pop()
|
||
G.add_edge(source, target)
|
||
repeated_nodes.append(target)
|
||
count = count + 1
|
||
|
||
repeated_nodes.extend([source] * m) # add source node to list m times
|
||
source += 1
|
||
return G
|
||
|
||
|
||
@py_random_state(3)
|
||
def random_lobster(n, p1, p2, seed=None):
|
||
"""Returns a random lobster graph.
|
||
|
||
A lobster is a tree that reduces to a caterpillar when pruning all
|
||
leaf nodes. A caterpillar is a tree that reduces to a path graph
|
||
when pruning all leaf nodes; setting `p2` to zero produces a caterpillar.
|
||
|
||
This implementation iterates on the probabilities `p1` and `p2` to add
|
||
edges at levels 1 and 2, respectively. Graphs are therefore constructed
|
||
iteratively with uniform randomness at each level rather than being selected
|
||
uniformly at random from the set of all possible lobsters.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
The expected number of nodes in the backbone
|
||
p1 : float
|
||
Probability of adding an edge to the backbone
|
||
p2 : float
|
||
Probability of adding an edge one level beyond backbone
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If `p1` or `p2` parameters are >= 1 because the while loops would never finish.
|
||
"""
|
||
p1, p2 = abs(p1), abs(p2)
|
||
if any([p >= 1 for p in [p1, p2]]):
|
||
raise nx.NetworkXError("Probability values for `p1` and `p2` must both be < 1.")
|
||
|
||
# a necessary ingredient in any self-respecting graph library
|
||
llen = int(2 * seed.random() * n + 0.5)
|
||
L = path_graph(llen)
|
||
# build caterpillar: add edges to path graph with probability p1
|
||
current_node = llen - 1
|
||
for n in range(llen):
|
||
while seed.random() < p1: # add fuzzy caterpillar parts
|
||
current_node += 1
|
||
L.add_edge(n, current_node)
|
||
cat_node = current_node
|
||
while seed.random() < p2: # add crunchy lobster bits
|
||
current_node += 1
|
||
L.add_edge(cat_node, current_node)
|
||
return L # voila, un lobster!
|
||
|
||
|
||
@py_random_state(1)
|
||
def random_shell_graph(constructor, seed=None):
|
||
"""Returns a random shell graph for the constructor given.
|
||
|
||
Parameters
|
||
----------
|
||
constructor : list of three-tuples
|
||
Represents the parameters for a shell, starting at the center
|
||
shell. Each element of the list must be of the form `(n, m,
|
||
d)`, where `n` is the number of nodes in the shell, `m` is
|
||
the number of edges in the shell, and `d` is the ratio of
|
||
inter-shell (next) edges to intra-shell edges. If `d` is zero,
|
||
there will be no intra-shell edges, and if `d` is one there
|
||
will be all possible intra-shell edges.
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Examples
|
||
--------
|
||
>>> constructor = [(10, 20, 0.8), (20, 40, 0.8)]
|
||
>>> G = nx.random_shell_graph(constructor)
|
||
|
||
"""
|
||
G = empty_graph(0)
|
||
|
||
glist = []
|
||
intra_edges = []
|
||
nnodes = 0
|
||
# create gnm graphs for each shell
|
||
for (n, m, d) in constructor:
|
||
inter_edges = int(m * d)
|
||
intra_edges.append(m - inter_edges)
|
||
g = nx.convert_node_labels_to_integers(
|
||
gnm_random_graph(n, inter_edges, seed=seed), first_label=nnodes
|
||
)
|
||
glist.append(g)
|
||
nnodes += n
|
||
G = nx.operators.union(G, g)
|
||
|
||
# connect the shells randomly
|
||
for gi in range(len(glist) - 1):
|
||
nlist1 = list(glist[gi])
|
||
nlist2 = list(glist[gi + 1])
|
||
total_edges = intra_edges[gi]
|
||
edge_count = 0
|
||
while edge_count < total_edges:
|
||
u = seed.choice(nlist1)
|
||
v = seed.choice(nlist2)
|
||
if u == v or G.has_edge(u, v):
|
||
continue
|
||
else:
|
||
G.add_edge(u, v)
|
||
edge_count = edge_count + 1
|
||
return G
|
||
|
||
|
||
@py_random_state(2)
|
||
def random_powerlaw_tree(n, gamma=3, seed=None, tries=100):
|
||
"""Returns a tree with a power law degree distribution.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
The number of nodes.
|
||
gamma : float
|
||
Exponent of the power law.
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
tries : int
|
||
Number of attempts to adjust the sequence to make it a tree.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If no valid sequence is found within the maximum number of
|
||
attempts.
|
||
|
||
Notes
|
||
-----
|
||
A trial power law degree sequence is chosen and then elements are
|
||
swapped with new elements from a powerlaw distribution until the
|
||
sequence makes a tree (by checking, for example, that the number of
|
||
edges is one smaller than the number of nodes).
|
||
|
||
"""
|
||
# This call may raise a NetworkXError if the number of tries is succeeded.
|
||
seq = random_powerlaw_tree_sequence(n, gamma=gamma, seed=seed, tries=tries)
|
||
G = degree_sequence_tree(seq)
|
||
return G
|
||
|
||
|
||
@py_random_state(2)
|
||
def random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100):
|
||
"""Returns a degree sequence for a tree with a power law distribution.
|
||
|
||
Parameters
|
||
----------
|
||
n : int,
|
||
The number of nodes.
|
||
gamma : float
|
||
Exponent of the power law.
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
tries : int
|
||
Number of attempts to adjust the sequence to make it a tree.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
If no valid sequence is found within the maximum number of
|
||
attempts.
|
||
|
||
Notes
|
||
-----
|
||
A trial power law degree sequence is chosen and then elements are
|
||
swapped with new elements from a power law distribution until
|
||
the sequence makes a tree (by checking, for example, that the number of
|
||
edges is one smaller than the number of nodes).
|
||
|
||
"""
|
||
# get trial sequence
|
||
z = nx.utils.powerlaw_sequence(n, exponent=gamma, seed=seed)
|
||
# round to integer values in the range [0,n]
|
||
zseq = [min(n, max(int(round(s)), 0)) for s in z]
|
||
|
||
# another sequence to swap values from
|
||
z = nx.utils.powerlaw_sequence(tries, exponent=gamma, seed=seed)
|
||
# round to integer values in the range [0,n]
|
||
swap = [min(n, max(int(round(s)), 0)) for s in z]
|
||
|
||
for deg in swap:
|
||
# If this degree sequence can be the degree sequence of a tree, return
|
||
# it. It can be a tree if the number of edges is one fewer than the
|
||
# number of nodes, or in other words, `n - sum(zseq) / 2 == 1`. We
|
||
# use an equivalent condition below that avoids floating point
|
||
# operations.
|
||
if 2 * n - sum(zseq) == 2:
|
||
return zseq
|
||
index = seed.randint(0, n - 1)
|
||
zseq[index] = swap.pop()
|
||
|
||
raise nx.NetworkXError(
|
||
f"Exceeded max ({tries}) attempts for a valid tree sequence."
|
||
)
|
||
|
||
|
||
@py_random_state(3)
|
||
def random_kernel_graph(n, kernel_integral, kernel_root=None, seed=None):
|
||
r"""Returns an random graph based on the specified kernel.
|
||
|
||
The algorithm chooses each of the $[n(n-1)]/2$ possible edges with
|
||
probability specified by a kernel $\kappa(x,y)$ [1]_. The kernel
|
||
$\kappa(x,y)$ must be a symmetric (in $x,y$), non-negative,
|
||
bounded function.
|
||
|
||
Parameters
|
||
----------
|
||
n : int
|
||
The number of nodes
|
||
kernal_integral : function
|
||
Function that returns the definite integral of the kernel $\kappa(x,y)$,
|
||
$F(y,a,b) := \int_a^b \kappa(x,y)dx$
|
||
kernel_root: function (optional)
|
||
Function that returns the root $b$ of the equation $F(y,a,b) = r$.
|
||
If None, the root is found using :func:`scipy.optimize.brentq`
|
||
(this requires SciPy).
|
||
seed : integer, random_state, or None (default)
|
||
Indicator of random number generation state.
|
||
See :ref:`Randomness<randomness>`.
|
||
|
||
Notes
|
||
-----
|
||
The kernel is specified through its definite integral which must be
|
||
provided as one of the arguments. If the integral and root of the
|
||
kernel integral can be found in $O(1)$ time then this algorithm runs in
|
||
time $O(n+m)$ where m is the expected number of edges [2]_.
|
||
|
||
The nodes are set to integers from $0$ to $n-1$.
|
||
|
||
Examples
|
||
--------
|
||
Generate an Erdős–Rényi random graph $G(n,c/n)$, with kernel
|
||
$\kappa(x,y)=c$ where $c$ is the mean expected degree.
|
||
|
||
>>> def integral(u, w, z):
|
||
... return c * (z - w)
|
||
>>> def root(u, w, r):
|
||
... return r / c + w
|
||
>>> c = 1
|
||
>>> graph = nx.random_kernel_graph(1000, integral, root)
|
||
|
||
See Also
|
||
--------
|
||
gnp_random_graph
|
||
expected_degree_graph
|
||
|
||
References
|
||
----------
|
||
.. [1] Bollobás, Béla, Janson, S. and Riordan, O.
|
||
"The phase transition in inhomogeneous random graphs",
|
||
*Random Structures Algorithms*, 31, 3--122, 2007.
|
||
|
||
.. [2] Hagberg A, Lemons N (2015),
|
||
"Fast Generation of Sparse Random Kernel Graphs".
|
||
PLoS ONE 10(9): e0135177, 2015. doi:10.1371/journal.pone.0135177
|
||
"""
|
||
if kernel_root is None:
|
||
import scipy.optimize as optimize
|
||
|
||
def kernel_root(y, a, r):
|
||
def my_function(b):
|
||
return kernel_integral(y, a, b) - r
|
||
|
||
return optimize.brentq(my_function, a, 1)
|
||
|
||
graph = nx.Graph()
|
||
graph.add_nodes_from(range(n))
|
||
(i, j) = (1, 1)
|
||
while i < n:
|
||
r = -math.log(1 - seed.random()) # (1-seed.random()) in (0, 1]
|
||
if kernel_integral(i / n, j / n, 1) <= r:
|
||
i, j = i + 1, i + 1
|
||
else:
|
||
j = int(math.ceil(n * kernel_root(i / n, j / n, r)))
|
||
graph.add_edge(i - 1, j - 1)
|
||
return graph
|