798 lines
29 KiB
Python
798 lines
29 KiB
Python
# -*- coding: utf-8 -*-
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# Copyright (C) 2004-2019 by
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# Aric Hagberg <hagberg@lanl.gov>
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# Dan Schult <dschult@colgate.edu>
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# Pieter Swart <swart@lanl.gov>
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# All rights reserved.
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# BSD license.
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#
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# Authors: Aric Hagberg (hagberg@lanl.gov)
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# Dan Schult (dschult@colgate.edu)
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# Ben Edwards (BJEdwards@gmail.com)
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# Arya McCarthy (admccarthy@smu.edu)
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# Cole MacLean (maclean.cole@gmail.com)
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"""Generators for geometric graphs.
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"""
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from bisect import bisect_left
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from itertools import combinations
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from itertools import product
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from math import sqrt
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import math
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try:
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from scipy.spatial import cKDTree as KDTree
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except ImportError:
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_is_scipy_available = False
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else:
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_is_scipy_available = True
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import networkx as nx
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from networkx.utils import nodes_or_number, py_random_state
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__all__ = ['geographical_threshold_graph', 'waxman_graph',
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'navigable_small_world_graph', 'random_geometric_graph',
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'soft_random_geometric_graph', 'thresholded_random_geometric_graph']
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def euclidean(x, y):
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"""Returns the Euclidean distance between the vectors ``x`` and ``y``.
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Each of ``x`` and ``y`` can be any iterable of numbers. The
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iterables must be of the same length.
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"""
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return sqrt(sum((a - b) ** 2 for a, b in zip(x, y)))
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def _fast_edges(G, radius, p):
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"""Returns edge list of node pairs within `radius` of each other
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using scipy KDTree and Minkowski distance metric `p`
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Requires scipy to be installed.
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"""
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pos = nx.get_node_attributes(G, 'pos')
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nodes, coords = list(zip(*pos.items()))
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kdtree = KDTree(coords) # Cannot provide generator.
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edge_indexes = kdtree.query_pairs(radius, p)
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edges = ((nodes[u], nodes[v]) for u, v in edge_indexes)
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return edges
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def _slow_edges(G, radius, p):
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"""Returns edge list of node pairs within `radius` of each other
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using Minkowski distance metric `p`
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Works without scipy, but in `O(n^2)` time.
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"""
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# TODO This can be parallelized.
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edges = []
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for (u, pu), (v, pv) in combinations(G.nodes(data='pos'), 2):
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if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius ** p:
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edges.append((u, v))
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return edges
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@py_random_state(5)
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@nodes_or_number(0)
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def random_geometric_graph(n, radius, dim=2, pos=None, p=2, seed=None):
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"""Returns a random geometric graph in the unit cube of dimensions `dim`.
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The random geometric graph model places `n` nodes uniformly at
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random in the unit cube. Two nodes are joined by an edge if the
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distance between the nodes is at most `radius`.
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Edges are determined using a KDTree when SciPy is available.
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This reduces the time complexity from $O(n^2)$ to $O(n)$.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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radius: float
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Distance threshold value
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dim : int, optional
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Dimension of graph
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pos : dict, optional
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A dictionary keyed by node with node positions as values.
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p : float, optional
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Which Minkowski distance metric to use. `p` has to meet the condition
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``1 <= p <= infinity``.
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If this argument is not specified, the :math:`L^2` metric
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(the Euclidean distance metric), p = 2 is used.
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This should not be confused with the `p` of an Erdős-Rényi random
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graph, which represents probability.
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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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Graph
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A random geometric graph, undirected and without self-loops.
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Each node has a node attribute ``'pos'`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function.
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Examples
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--------
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Create a random geometric graph on twenty nodes where nodes are joined by
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an edge if their distance is at most 0.1::
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>>> G = nx.random_geometric_graph(20, 0.1)
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Notes
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-----
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This uses a *k*-d tree to build the graph.
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The `pos` keyword argument can be used to specify node positions so you
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can create an arbitrary distribution and domain for positions.
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For example, to use a 2D Gaussian distribution of node positions with mean
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(0, 0) and standard deviation 2::
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>>> import random
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>>> n = 20
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>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
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>>> G = nx.random_geometric_graph(n, 0.2, pos=pos)
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References
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----------
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.. [1] Penrose, Mathew, *Random Geometric Graphs*,
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Oxford Studies in Probability, 5, 2003.
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"""
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# TODO Is this function just a special case of the geographical
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# threshold graph?
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#
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# n_name, nodes = n
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# half_radius = {v: radius / 2 for v in nodes}
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# return geographical_threshold_graph(nodes, theta=1, alpha=1,
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# weight=half_radius)
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#
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n_name, nodes = n
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G = nx.Graph()
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G.add_nodes_from(nodes)
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in nodes}
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nx.set_node_attributes(G, pos, 'pos')
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if _is_scipy_available:
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edges = _fast_edges(G, radius, p)
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else:
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edges = _slow_edges(G, radius, p)
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G.add_edges_from(edges)
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return G
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@py_random_state(6)
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@nodes_or_number(0)
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def soft_random_geometric_graph(n, radius, dim=2, pos=None, p=2, p_dist=None,
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seed=None):
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r"""Returns a soft random geometric graph in the unit cube.
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The soft random geometric graph [1] model places `n` nodes uniformly at
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random in the unit cube in dimension `dim`. Two nodes of distance, `dist`,
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computed by the `p`-Minkowski distance metric are joined by an edge with
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probability `p_dist` if the computed distance metric value of the nodes
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is at most `radius`, otherwise they are not joined.
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Edges within `radius` of each other are determined using a KDTree when
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SciPy is available. This reduces the time complexity from :math:`O(n^2)`
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to :math:`O(n)`.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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radius: float
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Distance threshold value
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dim : int, optional
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Dimension of graph
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pos : dict, optional
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A dictionary keyed by node with node positions as values.
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p : float, optional
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Which Minkowski distance metric to use.
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`p` has to meet the condition ``1 <= p <= infinity``.
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If this argument is not specified, the :math:`L^2` metric
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(the Euclidean distance metric), p = 2 is used.
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This should not be confused with the `p` of an Erdős-Rényi random
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graph, which represents probability.
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p_dist : function, optional
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A probability density function computing the probability of
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connecting two nodes that are of distance, dist, computed by the
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Minkowski distance metric. The probability density function, `p_dist`,
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must be any function that takes the metric value as input
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and outputs a single probability value between 0-1. The scipy.stats
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package has many probability distribution functions implemented and
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tools for custom probability distribution definitions [2], and passing
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the .pdf method of scipy.stats distributions can be used here. If the
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probability function, `p_dist`, is not supplied, the default function
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is an exponential distribution with rate parameter :math:`\lambda=1`.
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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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Graph
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A soft random geometric graph, undirected and without self-loops.
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Each node has a node attribute ``'pos'`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function.
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Examples
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--------
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Default Graph:
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G = nx.soft_random_geometric_graph(50, 0.2)
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Custom Graph:
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Create a soft random geometric graph on 100 uniformly distributed nodes
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where nodes are joined by an edge with probability computed from an
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exponential distribution with rate parameter :math:`\lambda=1` if their
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Euclidean distance is at most 0.2.
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Notes
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-----
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This uses a *k*-d tree to build the graph.
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The `pos` keyword argument can be used to specify node positions so you
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can create an arbitrary distribution and domain for positions.
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For example, to use a 2D Gaussian distribution of node positions with mean
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(0, 0) and standard deviation 2
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The scipy.stats package can be used to define the probability distribution
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with the .pdf method used as `p_dist`.
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::
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>>> import random
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>>> import math
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>>> n = 100
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>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
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>>> p_dist = lambda dist : math.exp(-dist)
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>>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist)
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References
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----------
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.. [1] Penrose, Mathew D. "Connectivity of soft random geometric graphs."
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The Annals of Applied Probability 26.2 (2016): 986-1028.
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[2] scipy.stats -
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https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html
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"""
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n_name, nodes = n
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G = nx.Graph()
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G.name = 'soft_random_geometric_graph({}, {}, {})'.format(n, radius, dim)
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G.add_nodes_from(nodes)
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in nodes}
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nx.set_node_attributes(G, pos, 'pos')
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# if p_dist function not supplied the default function is an exponential
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# distribution with rate parameter :math:`\lambda=1`.
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if p_dist is None:
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def p_dist(dist):
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return math.exp(-dist)
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def should_join(pair):
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u, v = pair
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u_pos, v_pos = pos[u], pos[v]
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dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos)))**(1 / p)
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# Check if dist <= radius parameter. This check is redundant if scipy
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# is available and _fast_edges routine is used, but provides the
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# check in case scipy is not available and all edge combinations
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# need to be checked
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if dist <= radius:
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return seed.random() < p_dist(dist)
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else:
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return False
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if _is_scipy_available:
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edges = _fast_edges(G, radius, p)
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G.add_edges_from(filter(should_join, edges))
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else:
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G.add_edges_from(filter(should_join, combinations(G, 2)))
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return G
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@py_random_state(7)
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@nodes_or_number(0)
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def geographical_threshold_graph(n, theta, dim=2, pos=None, weight=None,
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metric=None, p_dist=None, seed=None):
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r"""Returns a geographical threshold graph.
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The geographical threshold graph model places $n$ nodes uniformly at
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random in a rectangular domain. Each node $u$ is assigned a weight
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$w_u$. Two nodes $u$ and $v$ are joined by an edge if
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.. math::
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(w_u + w_v)h(r) \ge \theta
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where `r` is the distance between `u` and `v`, h(r) is a probability of
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connection as a function of `r`, and :math:`\theta` as the threshold
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parameter. h(r) corresponds to the p_dist parameter.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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theta: float
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Threshold value
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dim : int, optional
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Dimension of graph
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pos : dict
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Node positions as a dictionary of tuples keyed by node.
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weight : dict
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Node weights as a dictionary of numbers keyed by node.
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metric : function
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A metric on vectors of numbers (represented as lists or
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tuples). This must be a function that accepts two lists (or
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tuples) as input and yields a number as output. The function
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must also satisfy the four requirements of a `metric`_.
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Specifically, if $d$ is the function and $x$, $y$,
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and $z$ are vectors in the graph, then $d$ must satisfy
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1. $d(x, y) \ge 0$,
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2. $d(x, y) = 0$ if and only if $x = y$,
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3. $d(x, y) = d(y, x)$,
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4. $d(x, z) \le d(x, y) + d(y, z)$.
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If this argument is not specified, the Euclidean distance metric is
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used.
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.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
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p_dist : function, optional
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A probability density function computing the probability of
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connecting two nodes that are of distance, r, computed by metric.
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The probability density function, `p_dist`, must
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be any function that takes the metric value as input
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and outputs a single probability value between 0-1.
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The scipy.stats package has many probability distribution functions
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implemented and tools for custom probability distribution
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definitions [2], and passing the .pdf method of scipy.stats
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distributions can be used here. If the probability
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function, `p_dist`, is not supplied, the default exponential function
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:math: `r^{-2}` is used.
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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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Graph
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A random geographic threshold graph, undirected and without
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self-loops.
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Each node has a node attribute ``pos`` that stores the
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position of that node in Euclidean space as provided by the
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``pos`` keyword argument or, if ``pos`` was not provided, as
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generated by this function. Similarly, each node has a node
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attribute ``weight`` that stores the weight of that node as
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provided or as generated.
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Examples
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--------
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Specify an alternate distance metric using the ``metric`` keyword
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argument. For example, to use the `taxicab metric`_ instead of the
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default `Euclidean metric`_::
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>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
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>>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist)
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.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
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.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
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Notes
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-----
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If weights are not specified they are assigned to nodes by drawing randomly
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from the exponential distribution with rate parameter $\lambda=1$.
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To specify weights from a different distribution, use the `weight` keyword
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argument::
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>>> import random
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>>> n = 20
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>>> w = {i: random.expovariate(5.0) for i in range(n)}
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>>> G = nx.geographical_threshold_graph(20, 50, weight=w)
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If node positions are not specified they are randomly assigned from the
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uniform distribution.
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References
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----------
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.. [1] Masuda, N., Miwa, H., Konno, N.:
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Geographical threshold graphs with small-world and scale-free
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properties.
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Physical Review E 71, 036108 (2005)
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.. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus,
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Giant component and connectivity in geographical threshold graphs,
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in Algorithms and Models for the Web-Graph (WAW 2007),
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Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007
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"""
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n_name, nodes = n
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G = nx.Graph()
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G.add_nodes_from(nodes)
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# If no weights are provided, choose them from an exponential
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# distribution.
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if weight is None:
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weight = {v: seed.expovariate(1) for v in G}
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# If no positions are provided, choose uniformly random vectors in
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# Euclidean space of the specified dimension.
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if pos is None:
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pos = {v: [seed.random() for i in range(dim)] for v in nodes}
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# If no distance metric is provided, use Euclidean distance.
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if metric is None:
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metric = euclidean
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nx.set_node_attributes(G, weight, 'weight')
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nx.set_node_attributes(G, pos, 'pos')
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# if p_dist is not supplied, use default r^-2
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if p_dist is None:
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def p_dist(r):
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return r**-2
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# Returns ``True`` if and only if the nodes whose attributes are
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# ``du`` and ``dv`` should be joined, according to the threshold
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# condition.
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def should_join(pair):
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u, v = pair
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u_pos, v_pos = pos[u], pos[v]
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u_weight, v_weight = weight[u], weight[v]
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return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta
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G.add_edges_from(filter(should_join, combinations(G, 2)))
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return G
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@py_random_state(6)
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@nodes_or_number(0)
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def waxman_graph(n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1),
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metric=None, seed=None):
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r"""Returns a Waxman random graph.
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The Waxman random graph model places `n` nodes uniformly at random
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in a rectangular domain. Each pair of nodes at distance `d` is
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joined by an edge with probability
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.. math::
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p = \beta \exp(-d / \alpha L).
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This function implements both Waxman models, using the `L` keyword
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argument.
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* Waxman-1: if `L` is not specified, it is set to be the maximum distance
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between any pair of nodes.
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* Waxman-2: if `L` is specified, the distance between a pair of nodes is
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chosen uniformly at random from the interval `[0, L]`.
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Parameters
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----------
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n : int or iterable
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Number of nodes or iterable of nodes
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beta: float
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Model parameter
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alpha: float
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Model parameter
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L : float, optional
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Maximum distance between nodes. If not specified, the actual distance
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is calculated.
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domain : four-tuple of numbers, optional
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Domain size, given as a tuple of the form `(x_min, y_min, x_max,
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y_max)`.
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metric : function
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A metric on vectors of numbers (represented as lists or
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tuples). This must be a function that accepts two lists (or
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tuples) as input and yields a number as output. The function
|
|
must also satisfy the four requirements of a `metric`_.
|
|
Specifically, if $d$ is the function and $x$, $y$,
|
|
and $z$ are vectors in the graph, then $d$ must satisfy
|
|
|
|
1. $d(x, y) \ge 0$,
|
|
2. $d(x, y) = 0$ if and only if $x = y$,
|
|
3. $d(x, y) = d(y, x)$,
|
|
4. $d(x, z) \le d(x, y) + d(y, z)$.
|
|
|
|
If this argument is not specified, the Euclidean distance metric is
|
|
used.
|
|
|
|
.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
Graph
|
|
A random Waxman graph, undirected and without self-loops. Each
|
|
node has a node attribute ``'pos'`` that stores the position of
|
|
that node in Euclidean space as generated by this function.
|
|
|
|
Examples
|
|
--------
|
|
Specify an alternate distance metric using the ``metric`` keyword
|
|
argument. For example, to use the "`taxicab metric`_" instead of the
|
|
default `Euclidean metric`_::
|
|
|
|
>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
|
|
>>> G = nx.waxman_graph(10, 0.5, 0.1, metric=dist)
|
|
|
|
.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
|
|
.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
|
|
|
|
Notes
|
|
-----
|
|
Starting in NetworkX 2.0 the parameters alpha and beta align with their
|
|
usual roles in the probability distribution. In earlier versions their
|
|
positions in the expression were reversed. Their position in the calling
|
|
sequence reversed as well to minimize backward incompatibility.
|
|
|
|
References
|
|
----------
|
|
.. [1] B. M. Waxman, *Routing of multipoint connections*.
|
|
IEEE J. Select. Areas Commun. 6(9),(1988) 1617--1622.
|
|
"""
|
|
n_name, nodes = n
|
|
G = nx.Graph()
|
|
G.add_nodes_from(nodes)
|
|
(xmin, ymin, xmax, ymax) = domain
|
|
# Each node gets a uniformly random position in the given rectangle.
|
|
pos = {v: (seed.uniform(xmin, xmax), seed.uniform(ymin, ymax)) for v in G}
|
|
nx.set_node_attributes(G, pos, 'pos')
|
|
# If no distance metric is provided, use Euclidean distance.
|
|
if metric is None:
|
|
metric = euclidean
|
|
# If the maximum distance L is not specified (that is, we are in the
|
|
# Waxman-1 model), then find the maximum distance between any pair
|
|
# of nodes.
|
|
#
|
|
# In the Waxman-1 model, join nodes randomly based on distance. In
|
|
# the Waxman-2 model, join randomly based on random l.
|
|
if L is None:
|
|
L = max(metric(x, y) for x, y in combinations(pos.values(), 2))
|
|
|
|
def dist(u, v): return metric(pos[u], pos[v])
|
|
else:
|
|
def dist(u, v): return seed.random() * L
|
|
|
|
# `pair` is the pair of nodes to decide whether to join.
|
|
def should_join(pair):
|
|
return seed.random() < beta * math.exp(-dist(*pair) / (alpha * L))
|
|
|
|
G.add_edges_from(filter(should_join, combinations(G, 2)))
|
|
return G
|
|
|
|
|
|
@py_random_state(5)
|
|
def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None):
|
|
r"""Returns a navigable small-world graph.
|
|
|
|
A navigable small-world graph is a directed grid with additional long-range
|
|
connections that are chosen randomly.
|
|
|
|
[...] we begin with a set of nodes [...] that are identified with the set
|
|
of lattice points in an $n \times n$ square,
|
|
$\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$,
|
|
and we define the *lattice distance* between two nodes $(i, j)$ and
|
|
$(k, l)$ to be the number of "lattice steps" separating them:
|
|
$d((i, j), (k, l)) = |k - i| + |l - j|$.
|
|
|
|
For a universal constant $p >= 1$, the node $u$ has a directed edge to
|
|
every other node within lattice distance $p$---these are its *local
|
|
contacts*. For universal constants $q >= 0$ and $r >= 0$ we also
|
|
construct directed edges from $u$ to $q$ other nodes (the *long-range
|
|
contacts*) using independent random trials; the $i$th directed edge from
|
|
$u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$.
|
|
|
|
-- [1]_
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
The length of one side of the lattice; the number of nodes in
|
|
the graph is therefore $n^2$.
|
|
p : int
|
|
The diameter of short range connections. Each node is joined with every
|
|
other node within this lattice distance.
|
|
q : int
|
|
The number of long-range connections for each node.
|
|
r : float
|
|
Exponent for decaying probability of connections. The probability of
|
|
connecting to a node at lattice distance $d$ is $1/d^r$.
|
|
dim : int
|
|
Dimension of grid
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
References
|
|
----------
|
|
.. [1] J. Kleinberg. The small-world phenomenon: An algorithmic
|
|
perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.
|
|
"""
|
|
if (p < 1):
|
|
raise nx.NetworkXException("p must be >= 1")
|
|
if (q < 0):
|
|
raise nx.NetworkXException("q must be >= 0")
|
|
if (r < 0):
|
|
raise nx.NetworkXException("r must be >= 1")
|
|
|
|
G = nx.DiGraph()
|
|
nodes = list(product(range(n), repeat=dim))
|
|
for p1 in nodes:
|
|
probs = [0]
|
|
for p2 in nodes:
|
|
if p1 == p2:
|
|
continue
|
|
d = sum((abs(b - a) for a, b in zip(p1, p2)))
|
|
if d <= p:
|
|
G.add_edge(p1, p2)
|
|
probs.append(d**-r)
|
|
cdf = list(nx.utils.accumulate(probs))
|
|
for _ in range(q):
|
|
target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))]
|
|
G.add_edge(p1, target)
|
|
return G
|
|
|
|
|
|
@py_random_state(7)
|
|
@nodes_or_number(0)
|
|
def thresholded_random_geometric_graph(n, radius, theta, dim=2,
|
|
pos=None, weight=None, p=2, seed=None):
|
|
r"""Returns a thresholded random geometric graph in the unit cube.
|
|
|
|
The thresholded random geometric graph [1] model places `n` nodes
|
|
uniformly at random in the unit cube of dimensions `dim`. Each node
|
|
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
|
|
joined by an edge if they are within the maximum connection distance,
|
|
`radius` computed by the `p`-Minkowski distance and the summation of
|
|
weights :math:`w_u` + :math:`w_v` is greater than or equal
|
|
to the threshold parameter `theta`.
|
|
|
|
Edges within `radius` of each other are determined using a KDTree when
|
|
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
|
|
to :math:`O(n)`.
|
|
|
|
Parameters
|
|
----------
|
|
n : int or iterable
|
|
Number of nodes or iterable of nodes
|
|
radius: float
|
|
Distance threshold value
|
|
theta: float
|
|
Threshold value
|
|
dim : int, optional
|
|
Dimension of graph
|
|
pos : dict, optional
|
|
A dictionary keyed by node with node positions as values.
|
|
weight : dict, optional
|
|
Node weights as a dictionary of numbers keyed by node.
|
|
p : float, optional
|
|
Which Minkowski distance metric to use. `p` has to meet the condition
|
|
``1 <= p <= infinity``.
|
|
|
|
If this argument is not specified, the :math:`L^2` metric
|
|
(the Euclidean distance metric), p = 2 is used.
|
|
|
|
This should not be confused with the `p` of an Erdős-Rényi random
|
|
graph, which represents probability.
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
Graph
|
|
A thresholded random geographic graph, undirected and without
|
|
self-loops.
|
|
|
|
Each node has a node attribute ``'pos'`` that stores the
|
|
position of that node in Euclidean space as provided by the
|
|
``pos`` keyword argument or, if ``pos`` was not provided, as
|
|
generated by this function. Similarly, each node has a nodethre
|
|
attribute ``'weight'`` that stores the weight of that node as
|
|
provided or as generated.
|
|
|
|
Examples
|
|
--------
|
|
Default Graph:
|
|
|
|
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
|
|
|
|
Custom Graph:
|
|
|
|
Create a thresholded random geometric graph on 50 uniformly distributed
|
|
nodes where nodes are joined by an edge if their sum weights drawn from
|
|
a exponential distribution with rate = 5 are >= theta = 0.1 and their
|
|
Euclidean distance is at most 0.2.
|
|
|
|
Notes
|
|
-----
|
|
This uses a *k*-d tree to build the graph.
|
|
|
|
The `pos` keyword argument can be used to specify node positions so you
|
|
can create an arbitrary distribution and domain for positions.
|
|
|
|
For example, to use a 2D Gaussian distribution of node positions with mean
|
|
(0, 0) and standard deviation 2
|
|
|
|
If weights are not specified they are assigned to nodes by drawing randomly
|
|
from the exponential distribution with rate parameter :math:`\lambda=1`.
|
|
To specify weights from a different distribution, use the `weight` keyword
|
|
argument::
|
|
|
|
::
|
|
|
|
>>> import random
|
|
>>> import math
|
|
>>> n = 50
|
|
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
|
|
>>> w = {i: random.expovariate(5.0) for i in range(n)}
|
|
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
|
|
|
|
References
|
|
----------
|
|
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
|
|
|
|
"""
|
|
|
|
n_name, nodes = n
|
|
G = nx.Graph()
|
|
namestr = 'thresholded_random_geometric_graph({}, {}, {}, {})'
|
|
G.name = namestr.format(n, radius, theta, dim)
|
|
G.add_nodes_from(nodes)
|
|
# If no weights are provided, choose them from an exponential
|
|
# distribution.
|
|
if weight is None:
|
|
weight = {v: seed.expovariate(1) for v in G}
|
|
# If no positions are provided, choose uniformly random vectors in
|
|
# Euclidean space of the specified dimension.
|
|
if pos is None:
|
|
pos = {v: [seed.random() for i in range(dim)] for v in nodes}
|
|
# If no distance metric is provided, use Euclidean distance.
|
|
|
|
nx.set_node_attributes(G, weight, 'weight')
|
|
nx.set_node_attributes(G, pos, 'pos')
|
|
|
|
# Returns ``True`` if and only if the nodes whose attributes are
|
|
# ``du`` and ``dv`` should be joined, according to the threshold
|
|
# condition and node pairs are within the maximum connection
|
|
# distance, ``radius``.
|
|
def should_join(pair):
|
|
u, v = pair
|
|
u_weight, v_weight = weight[u], weight[v]
|
|
u_pos, v_pos = pos[u], pos[v]
|
|
dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos)))**(1 / p)
|
|
# Check if dist is <= radius parameter. This check is redundant if
|
|
# scipy is available and _fast_edges routine is used, but provides
|
|
# the check in case scipy is not available and all edge combinations
|
|
# need to be checked
|
|
if dist <= radius:
|
|
return theta <= u_weight + v_weight
|
|
else:
|
|
return False
|
|
|
|
if _is_scipy_available:
|
|
edges = _fast_edges(G, radius, p)
|
|
G.add_edges_from(filter(should_join, edges))
|
|
else:
|
|
G.add_edges_from(filter(should_join, combinations(G, 2)))
|
|
|
|
return G
|