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## iof-tools / networkxMiCe / networkx-master / networkx / generators / geometric.py @ 5cef0f13

 1 # -*- coding: utf-8 -*-  # Copyright (C) 2004-2019 by  # Aric Hagberg  # Dan Schult  # Pieter Swart  # All rights reserved.  # BSD license.  #  # Authors: Aric Hagberg (hagberg@lanl.gov)  # Dan Schult (dschult@colgate.edu)  # Ben Edwards (BJEdwards@gmail.com)  # Arya McCarthy (admccarthy@smu.edu)  # Cole MacLean (maclean.cole@gmail.com)  """Generators for geometric graphs.  """  from __future__ import division  from bisect import bisect_left  from itertools import combinations  from itertools import product  from math import sqrt  import math  try:   from scipy.spatial import cKDTree as KDTree  except ImportError:   _is_scipy_available = False  else:   _is_scipy_available = True  import networkx as nx  from networkx.utils import nodes_or_number, py_random_state  __all__ = ['geographical_threshold_graph', 'waxman_graph',   'navigable_small_world_graph', 'random_geometric_graph',   'soft_random_geometric_graph', 'thresholded_random_geometric_graph']  def euclidean(x, y):   """Returns the Euclidean distance between the vectors x and y.     Each of x and y can be any iterable of numbers. The   iterables must be of the same length.     """   return sqrt(sum((a - b) ** 2 for a, b in zip(x, y)))  def _fast_edges(G, radius, p):   """Returns edge list of node pairs within radius of each other   using scipy KDTree and Minkowski distance metric p     Requires scipy to be installed.   """   pos = nx.get_node_attributes(G, 'pos')   nodes, coords = list(zip(*pos.items()))   kdtree = KDTree(coords) # Cannot provide generator.   edge_indexes = kdtree.query_pairs(radius, p)   edges = ((nodes[u], nodes[v]) for u, v in edge_indexes)   return edges  def _slow_edges(G, radius, p):   """Returns edge list of node pairs within radius of each other   using Minkowski distance metric p     Works without scipy, but in O(n^2) time.   """   # TODO This can be parallelized.   edges = []   for (u, pu), (v, pv) in combinations(G.nodes(data='pos'), 2):   if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius ** p:   edges.append((u, v))   return edges  @py_random_state(5)  @nodes_or_number(0)  def random_geometric_graph(n, radius, dim=2, pos=None, p=2, seed=None):   """Returns a random geometric graph in the unit cube of dimensions dim.     The random geometric graph model places n nodes uniformly at   random in the unit cube. Two nodes are joined by an edge if the   distance between the nodes is at most radius.     Edges are determined using a KDTree when SciPy is available.   This reduces the time complexity from $O(n^2)$ to $O(n)$.     Parameters   ----------   n : int or iterable   Number of nodes or iterable of nodes   radius: float   Distance threshold value   dim : int, optional   Dimension of graph   pos : dict, optional   A dictionary keyed by node with node positions as values.   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.     Returns   -------   Graph   A random geometric 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.     Examples   --------   Create a random geometric graph on twenty nodes where nodes are joined by   an edge if their distance is at most 0.1::     >>> G = nx.random_geometric_graph(20, 0.1)     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::     >>> import random   >>> n = 20   >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}   >>> G = nx.random_geometric_graph(n, 0.2, pos=pos)     References   ----------   .. [1] Penrose, Mathew, *Random Geometric Graphs*,   Oxford Studies in Probability, 5, 2003.     """   # TODO Is this function just a special case of the geographical   # threshold graph?   #   # n_name, nodes = n   # half_radius = {v: radius / 2 for v in nodes}   # return geographical_threshold_graph(nodes, theta=1, alpha=1,   # weight=half_radius)   #   n_name, nodes = n   G = nx.Graph()   G.add_nodes_from(nodes)   # 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}   nx.set_node_attributes(G, pos, 'pos')   if _is_scipy_available:   edges = _fast_edges(G, radius, p)   else:   edges = _slow_edges(G, radius, p)   G.add_edges_from(edges)   return G  @py_random_state(6)  @nodes_or_number(0)  def soft_random_geometric_graph(n, radius, dim=2, pos=None, p=2, p_dist=None,   seed=None):   r"""Returns a soft random geometric graph in the unit cube.     The soft random geometric graph [1] model places n nodes uniformly at   random in the unit cube in dimension dim. Two nodes of distance, dist,   computed by the p-Minkowski distance metric are joined by an edge with   probability p_dist if the computed distance metric value of the nodes   is at most radius, otherwise they are not joined.     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   dim : int, optional   Dimension of graph   pos : dict, optional   A dictionary keyed by node with node positions as values.   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.   p_dist : function, optional   A probability density function computing the probability of   connecting two nodes that are of distance, dist, computed by the   Minkowski distance metric. The probability density function, p_dist,   must be any function that takes the metric value as input   and outputs a single probability value between 0-1. The scipy.stats   package has many probability distribution functions implemented and   tools for custom probability distribution definitions [2], and passing   the .pdf method of scipy.stats distributions can be used here. If the   probability function, p_dist, is not supplied, the default function   is an exponential distribution with rate parameter :math:\lambda=1.   seed : integer, random_state, or None (default)   Indicator of random number generation state.   See :ref:Randomness.     Returns   -------   Graph   A soft random geometric 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.     Examples   --------   Default Graph:     G = nx.soft_random_geometric_graph(50, 0.2)     Custom Graph:     Create a soft random geometric graph on 100 uniformly distributed nodes   where nodes are joined by an edge with probability computed from an   exponential distribution with rate parameter :math:\lambda=1 if 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     The scipy.stats package can be used to define the probability distribution   with the .pdf method used as p_dist.     ::     >>> import random   >>> import math   >>> n = 100   >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}   >>> p_dist = lambda dist : math.exp(-dist)   >>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist)     References   ----------   .. [1] Penrose, Mathew D. "Connectivity of soft random geometric graphs."   The Annals of Applied Probability 26.2 (2016): 986-1028.   [2] scipy.stats -   https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html     """   n_name, nodes = n   G = nx.Graph()   G.name = 'soft_random_geometric_graph({}, {}, {})'.format(n, radius, dim)   G.add_nodes_from(nodes)   # 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}   nx.set_node_attributes(G, pos, 'pos')   # if p_dist function not supplied the default function is an exponential   # distribution with rate parameter :math:\lambda=1.   if p_dist is None:   def p_dist(dist):   return math.exp(-dist)   def should_join(pair):   u, v = pair   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 <= 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 seed.random() < p_dist(dist)   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  @py_random_state(7)  @nodes_or_number(0)  def geographical_threshold_graph(n, theta, dim=2, pos=None, weight=None,   metric=None, p_dist=None, seed=None):   r"""Returns a geographical threshold graph.     The geographical threshold graph model places $n$ nodes uniformly at   random in a rectangular domain. Each node $u$ is assigned a weight   $w_u$. Two nodes $u$ and $v$ are joined by an edge if     .. math::     (w_u + w_v)h(r) \ge \theta     where r is the distance between u and v, h(r) is a probability of   connection as a function of r, and :math:\theta as the threshold   parameter. h(r) corresponds to the p_dist parameter.     Parameters   ----------   n : int or iterable   Number of nodes or iterable of nodes   theta: float   Threshold value   dim : int, optional   Dimension of graph   pos : dict   Node positions as a dictionary of tuples keyed by node.   weight : dict   Node weights as a dictionary of numbers keyed by node.   metric : function   A metric on vectors of numbers (represented as lists or   tuples). This must be a function that accepts two lists (or   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   p_dist : function, optional   A probability density function computing the probability of   connecting two nodes that are of distance, r, computed by metric.   The probability density function, p_dist, must   be any function that takes the metric value as input   and outputs a single probability value between 0-1.   The scipy.stats package has many probability distribution functions   implemented and tools for custom probability distribution   definitions [2], and passing the .pdf method of scipy.stats   distributions can be used here. If the probability   function, p_dist, is not supplied, the default exponential function   :math: r^{-2} is used.   seed : integer, random_state, or None (default)   Indicator of random number generation state.   See :ref:Randomness.     Returns   -------   Graph   A random geographic threshold 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 node   attribute weight that stores the weight of that node as   provided or as generated.     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.geographical_threshold_graph(10, 0.1, metric=dist)     .. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry   .. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance     Notes   -----   If weights are not specified they are assigned to nodes by drawing randomly   from the exponential distribution with rate parameter $\lambda=1$.   To specify weights from a different distribution, use the weight keyword   argument::     >>> import random   >>> n = 20   >>> w = {i: random.expovariate(5.0) for i in range(n)}   >>> G = nx.geographical_threshold_graph(20, 50, weight=w)     If node positions are not specified they are randomly assigned from the   uniform distribution.     Starting in NetworkX 2.1 the parameter alpha is deprecated and replaced   with the customizable p_dist function parameter, which defaults to r^-2   if p_dist is not supplied. To reproduce networks of earlier NetworkX   versions, a custom function needs to be defined and passed as the   p_dist parameter. For example, if the parameter alpha = 2 was used   in NetworkX 2.0, the custom function def custom_dist(r): r**-2 can be   passed in versions >=2.1 as the parameter p_dist = custom_dist to   produce an equivalent network. Note the change in sign from +2 to -2 in   this parameter change.     References   ----------   .. [1] Masuda, N., Miwa, H., Konno, N.:   Geographical threshold graphs with small-world and scale-free   properties.   Physical Review E 71, 036108 (2005)   .. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus,   Giant component and connectivity in geographical threshold graphs,   in Algorithms and Models for the Web-Graph (WAW 2007),   Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007   """   n_name, nodes = n   G = nx.Graph()   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.   if metric is None:   metric = euclidean   nx.set_node_attributes(G, weight, 'weight')   nx.set_node_attributes(G, pos, 'pos')   # if p_dist is not supplied, use default r^-2   if p_dist is None:   def p_dist(r):   return r**-2   # Returns True if and only if the nodes whose attributes are   # du and dv should be joined, according to the threshold   # condition.   def should_join(pair):   u, v = pair   u_pos, v_pos = pos[u], pos[v]   u_weight, v_weight = weight[u], weight[v]   return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta   G.add_edges_from(filter(should_join, combinations(G, 2)))   return G  @py_random_state(6)  @nodes_or_number(0)  def waxman_graph(n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1),   metric=None, seed=None):   r"""Returns a Waxman random graph.     The Waxman random graph model places n nodes uniformly at random   in a rectangular domain. Each pair of nodes at distance d is   joined by an edge with probability     .. math::   p = \beta \exp(-d / \alpha L).     This function implements both Waxman models, using the L keyword   argument.     * Waxman-1: if L is not specified, it is set to be the maximum distance   between any pair of nodes.   * Waxman-2: if L is specified, the distance between a pair of nodes is   chosen uniformly at random from the interval [0, L].     Parameters   ----------   n : int or iterable   Number of nodes or iterable of nodes   beta: float   Model parameter   alpha: float   Model parameter   L : float, optional   Maximum distance between nodes. If not specified, the actual distance   is calculated.   domain : four-tuple of numbers, optional   Domain size, given as a tuple of the form (x_min, y_min, x_max,   y_max).   metric : function   A metric on vectors of numbers (represented as lists or   tuples). This must be a function that accepts two lists (or   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.     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.     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.     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