Source code for networkx.generators.geometric

# -*- coding: utf-8 -*-
"""
Generators for geometric graphs.
"""
#    Copyright (C) 2004-2015 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.

__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):
    """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 Euclidean distance
    between the nodes is at most ``radius``.

    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
    --------
    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 algorithm currently only supports Euclidean distance.

    This uses an `O(n^2)` algorithm to build the graph.  A faster algorithm
    is possible using k-d trees.

    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.
    """
    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"""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 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 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
    """
    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):
    """Generates 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 model places ``n`` nodes uniformly at random in a
    rectangular domain. Each pair of nodes at Euclidean distance `d` is joined
    by an edge with probability

    .. math::
            p = \alpha \exp(-d / \beta 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
        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 : four-tuple of numbers, optional
        Domain size, given as a tuple of the form `(x_min, y_min, x_max,
        y_max)`.

    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):
    """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 \geq 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 \ge 0` and `r \ge 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 number of nodes.
    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 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.accumulate(probs))
        for _ in range(q):
            target = nodes[bisect_left(cdf,random.uniform(0, cdf[-1]))]
            G.add_edge(p1,target)
    return G