# -*- coding: utf-8 -*-
# Copyright (C) 2004-2017 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# 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)
"""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
import random
from random import uniform
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
__all__ = ['geographical_threshold_graph', 'waxman_graph',
'navigable_small_world_graph', '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_construct_edges(G, radius, p):
"""Construct edges for random geometric graph.
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)
G.add_edges_from(edges)
def _slow_construct_edges(G, radius, p):
"""Construct edges for random geometric graph.
Works without scipy, but in `O(n^2)` time.
"""
# TODO This can be parallelized.
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:
G.add_edge(u, v)
[docs]@nodes_or_number(0)
def random_geometric_graph(n, radius, dim=2, pos=None, p=2):
"""Returns a random geometric graph in the unit cube.
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
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the $L^2$ metric
(the Euclidean distance metric) is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
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
>>> p = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> G = nx.random_geometric_graph(n, 0.2, pos=p)
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: [random.random() for i in range(dim)] for v in nodes}
nx.set_node_attributes(G, pos, 'pos')
if _is_scipy_available:
_fast_construct_edges(G, radius, p)
else:
_slow_construct_edges(G, radius, p)
return G
[docs]@nodes_or_number(0)
def geographical_threshold_graph(n, theta, alpha=2, dim=2, pos=None,
weight=None, metric=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 \ge \theta r^{\alpha}
where $r$ is the distance between $u$ and $v$, and $\theta$,
$\alpha$ are parameters.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
theta: float
Threshold value
alpha: float, optional
Exponent of distance function
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
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.
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: random.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: [random.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')
# 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 theta * metric(u_pos, v_pos) ** alpha <= u_weight + v_weight
G.add_edges_from(filter(should_join, combinations(G, 2)))
return G
[docs]@nodes_or_number(0)
def waxman_graph(n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1),
metric=None):
r"""Return 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
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 expresssion 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: (uniform(xmin, xmax), 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))
dist = lambda u, v: metric(pos[u], pos[v])
else:
dist = lambda u, v: random.random() * L
# `pair` is the pair of nodes to decide whether to join.
def should_join(pair):
return random.random() < beta * math.exp(-dist(*pair) / (alpha * L))
G.add_edges_from(filter(should_join, combinations(G, 2)))
return G
[docs]def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None):
r"""Return 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 : int, optional
Seed for random number generator (default=None).
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")
if seed is not None:
random.seed(seed)
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, random.uniform(0, cdf[-1]))]
G.add_edge(p1, target)
return G