# -*- coding: utf-8 -*-
"""
Generators for geometric graphs.
"""
# Copyright (C) 2004-2011 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
from __future__ import print_function
__author__ = "\n".join(['Aric Hagberg (hagberg@lanl.gov)',
'Dan Schult (dschult@colgate.edu)',
'Ben Edwards (BJEdwards@gmail.com)'])
__all__ = ['random_geometric_graph',
'waxman_graph',
'geographical_threshold_graph',
'navigable_small_world_graph']
from bisect import bisect_left
from functools import reduce
from itertools import product
import math, random, sys
import networkx as nx
#---------------------------------------------------------------------------
# Random Geometric Graphs
#---------------------------------------------------------------------------
[docs]def random_geometric_graph(n, radius, dim=2, pos=None):
r"""Return the random geometric graph in the unit cube.
The random geometric graph model places n nodes uniformly at random
in the unit cube Two nodes `u,v` are connected with an edge if
`d(u,v)<=r` where `d` is the Euclidean distance and `r` is a radius
threshold.
Parameters
----------
n : int
Number 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.
Returns
-------
Graph
Examples
--------
>>> G = nx.random_geometric_graph(20,0.1)
Notes
-----
This uses an `n^2` algorithm to build the graph. A faster algorithm
is possible using k-d trees.
The pos keyword can be used to specify node positions so you can create
an arbitrary distribution and domain for positions. If you need a distance
function other than Euclidean you'll have to hack the algorithm.
E.g to use a 2d Gaussian distribution of node positions with mean (0,0)
and std. dev. 2
>>> import random
>>> n=20
>>> p=dict((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.
"""
G=nx.Graph()
G.name="Random Geometric Graph"
G.add_nodes_from(range(n))
if pos is None:
# random positions
for n in G:
G.node[n]['pos']=[random.random() for i in range(0,dim)]
else:
nx.set_node_attributes(G,'pos',pos)
# connect nodes within "radius" of each other
# n^2 algorithm, could use a k-d tree implementation
nodes = G.nodes(data=True)
while nodes:
u,du = nodes.pop()
pu = du['pos']
for v,dv in nodes:
pv = dv['pos']
d = sum(((a-b)**2 for a,b in zip(pu,pv)))
if d <= radius**2:
G.add_edge(u,v)
return G
[docs]def geographical_threshold_graph(n, theta, alpha=2, dim=2,
pos=None, weight=None):
r"""Return 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,v` are connected with an edge if
.. math::
w_u + w_v \ge \theta r^{\alpha}
where `r` is the Euclidean distance between `u` and `v`,
and `\theta`, `\alpha` are parameters.
Parameters
----------
n : int
Number 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.
Returns
-------
Graph
Examples
--------
>>> G = nx.geographical_threshold_graph(20,50)
Notes
-----
If weights are not specified they are assigned to nodes by drawing randomly
from an the exponential distribution with rate parameter `\lambda=1`.
To specify a weights from a different distribution assign them to a
dictionary and pass it as the weight= keyword
>>> import random
>>> n = 20
>>> w=dict((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
"""
G=nx.Graph()
# add n nodes
G.add_nodes_from([v for v in range(n)])
if weight is None:
# choose weights from exponential distribution
for n in G:
G.node[n]['weight'] = random.expovariate(1.0)
else:
nx.set_node_attributes(G,'weight',weight)
if pos is None:
# random positions
for n in G:
G.node[n]['pos']=[random.random() for i in range(0,dim)]
else:
nx.set_node_attributes(G,'pos',pos)
G.add_edges_from(geographical_threshold_edges(G, theta, alpha))
return G
def geographical_threshold_edges(G, theta, alpha=2):
# generate edges for a geographical threshold graph given a graph
# with positions and weights assigned as node attributes 'pos' and 'weight'.
nodes = G.nodes(data=True)
while nodes:
u,du = nodes.pop()
wu = du['weight']
pu = du['pos']
for v,dv in nodes:
wv = dv['weight']
pv = dv['pos']
r = math.sqrt(sum(((a-b)**2 for a,b in zip(pu,pv))))
if wu+wv >= theta*r**alpha:
yield(u,v)
[docs]def waxman_graph(n, alpha=0.4, beta=0.1, L=None, domain=(0,0,1,1)):
r"""Return a Waxman random graph.
The Waxman random graph models place n nodes uniformly at random
in a rectangular domain. Two nodes u,v are connected with an edge
with probability
.. math::
p = \alpha*exp(d/(\beta*L)).
This function implements both Waxman models.
Waxman-1: `L` not specified
The distance `d` is the Euclidean distance between the nodes u and v.
`L` is the maximum distance between all nodes in the graph.
Waxman-2: `L` specified
The distance `d` is chosen randomly in `[0,L]`.
Parameters
----------
n : int
Number of nodes
alpha: float
Model parameter
beta: float
Model parameter
L : float, optional
Maximum distance between nodes. If not specified the actual distance
is calculated.
domain : tuple of numbers, optional
Domain size (xmin, ymin, xmax, ymax)
Returns
-------
G: Graph
References
----------
.. [1] B. M. Waxman, Routing of multipoint connections.
IEEE J. Select. Areas Commun. 6(9),(1988) 1617-1622.
"""
# build graph of n nodes with random positions in the unit square
G = nx.Graph()
G.add_nodes_from(range(n))
(xmin,ymin,xmax,ymax)=domain
for n in G:
G.node[n]['pos']=((xmin + (xmax-xmin))*random.random(),
(ymin + (ymax-ymin))*random.random())
if L is None:
# find maximum distance L between two nodes
l = 0
pos = list(nx.get_node_attributes(G,'pos').values())
while pos:
x1,y1 = pos.pop()
for x2,y2 in pos:
r2 = (x1-x2)**2 + (y1-y2)**2
if r2 > l:
l = r2
l=math.sqrt(l)
else:
# user specified maximum distance
l = L
nodes=G.nodes()
if L is None:
# Waxman-1 model
# try all pairs, connect randomly based on euclidean distance
while nodes:
u = nodes.pop()
x1,y1 = G.node[u]['pos']
for v in nodes:
x2,y2 = G.node[v]['pos']
r = math.sqrt((x1-x2)**2 + (y1-y2)**2)
if random.random() < alpha*math.exp(-r/(beta*l)):
G.add_edge(u,v)
else:
# Waxman-2 model
# try all pairs, connect randomly based on randomly chosen l
while nodes:
u = nodes.pop()
for v in nodes:
r = random.random()*l
if random.random() < alpha*math.exp(-r/(beta*l)):
G.add_edge(u,v)
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. From [1]_:
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 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`, the node `u` has a directed edge to every other
node within lattice distance `p` (local contacts) .
For universal constants `q\ge 0` and `r\ge 0` construct directed edges from `u` to `q`
other nodes (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}`.
Parameters
----------
n : int
The number of nodes.
p : int
The diameter of short range connections. Each node is connected
to every other node within lattice distance p.
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 not seed is 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.cumulative_sum(probs))
for _ in range(q):
target = nodes[bisect_left(cdf,random.uniform(0, cdf[-1]))]
G.add_edge(p1,target)
return G