NetworkX

Source code for networkx.generators.geometric

# -*- coding: utf-8 -*-
"""
Generators for geometric graphs.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>

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