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This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.generators.geometric

# -*- coding: utf-8 -*-
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#
# Authors: Aric Hagberg (hagberg@lanl.gov)
#          Dan Schult (dschult@colgate.edu)
#          Ben Edwards (BJEdwards@gmail.com)
#          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)))

"""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.
edges = ((nodes[u], nodes[v]) for u, v in edge_indexes)
return edges

"""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

[docs]@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<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
[docs]@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<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
[docs]@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<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
[docs]@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<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
[docs]@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