__init__.py 20.6 KB
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#! /usr/bin/env python
# graph_tool.py -- a general graph manipulation python module
#
# Copyright (C) 2007 Tiago de Paula Peixoto <tiago@forked.de>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

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"""
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``graph_tool.generation`` - Random graph generation
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---------------------------------------------------
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Summary
+++++++

.. autosummary::
   :nosignatures:

   random_graph
   random_rewire
   predecessor_tree
   line_graph
   graph_union
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   triangulation
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Contents
++++++++
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"""

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from .. dl_import import dl_import
dl_import("import libgraph_tool_generation")
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from .. core import Graph, _check_prop_scalar, _prop, _limit_args
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import sys, numpy, numpy.random
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__all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph",
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           "graph_union", "triangulation"]
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def _corr_wrap(i, j, corr):
    return corr(i[1], j[1])

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def random_graph(N, deg_sampler, deg_corr=None, directed=True,
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                 parallel=False, self_loops=False, verbose=False):
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    r"""
    Generate a random graph, with a given degree distribution and correlation.

    Parameters
    ----------
    N : int
        Number of vertices in the graph.
    deg_sampler : function
        A degree sampler function which is called without arguments, and returns
        a tuple of ints representing the in and out-degree of a given vertex (or
        a single int for undirected graphs, representing the out-degree). This
        function is called once per vertex, but may be called more times, if the
        degree sequence cannot be used to build a graph.
    deg_corr : function (optional, default: None)
        A function which give the degree correlation of the graph. It should be
        callable with two parameters: the in,out-degree pair of the source
        vertex an edge, and the in,out-degree pair of the target of the same
        edge (for undirected graphs, both parameters are single values). The
        function should return a number proportional to the probability of such
        an edge existing in the generated graph.
    directed : bool (optional, default: True)
        Whether the generated graph should be directed.
    parallel : bool (optional, default: False)
        If True, parallel edges are allowed.
    self_loops : bool (optional, default: False)
        If True, self-loops are allowed.

    Returns
    -------
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    random_graph : :class:`~graph_tool.Graph`
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        The generated graph.

    See Also
    --------
    random_rewire: in place graph shuffling

    Notes
    -----
    The algorithm maintains a list of all available source and target degree
    pairs, such that the deg_corr function is called only once with the same
    parameters.

    The uncorrelated case, the complexity is :math:`O(V+E)`. For the correlated
    case the worst-case complexity is :math:`O(V^2)`, but the typical case has
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    complexity :math:`O(V + E\log N_k + N_k^2)`, where :math:`N_k < V` is the
    number of different degrees sampled (or in,out-degree pairs).
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    Examples
    --------

    >>> from numpy.random import randint, random, seed, poisson
    >>> from pylab import *
    >>> seed(42)

    This is a degree sampler which uses rejection sampling to sample from the
    distribution :math:`P(k)\propto 1/k`, up to a maximum.

    >>> def sample_k(max):
    ...     accept = False
    ...     while not accept:
    ...         k = randint(1,max+1)
    ...         accept = random() < 1.0/k
    ...     return k
    ...

    The following generates a random undirected graph with degree distribution
    :math:`P(k)\propto 1/k` (with k_max=40) and an *assortative* degree
    correlation of the form:

    .. math::

        P(i,k) \propto \frac{1}{1+|i-k|}

    >>> g = gt.random_graph(1000, lambda: sample_k(40),
    ...                     lambda i,k: 1.0/(1+abs(i-k)), directed=False)
    >>> gt.scalar_assortativity(g, "out")
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    (0.60296352140954257, 0.011780362691333932)
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    The following samples an in,out-degree pair from the joint distribution:

    .. math::

        p(j,k) = \frac{1}{2}\frac{e^{-m_1}m_1^j}{j!}\frac{e^{-m_1}m_1^k}{k!} +
                 \frac{1}{2}\frac{e^{-m_2}m_2^j}{j!}\frac{e^{-m_2}m_2^k}{k!}

    with :math:`m_1 = 4` and :math:`m_2 = 20`.

    >>> def deg_sample():
    ...    if random() > 0.5:
    ...        return poisson(4), poisson(4)
    ...    else:
    ...        return poisson(20), poisson(20)
    ...

    The following generates a random directed graph with this distribution, and
    plots the combined degree correlation.

    >>> g = gt.random_graph(20000, deg_sample)
    >>>
    >>> hist = gt.combined_corr_hist(g, "in", "out")
    >>> imshow(hist[0], interpolation="nearest")
    <...>
    >>> colorbar()
    <...>
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    >>> xlabel("in-degree")
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    <...>
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    >>> ylabel("out-degree")
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    <...>
    >>> savefig("combined-deg-hist.png")

    .. figure:: combined-deg-hist.png
        :align: center

        Combined degree histogram.

    A correlated directed graph can be build as follows. Consider the following
    degree correlation:

    .. math::

         P(j',k'|j,k)=\frac{e^{-k}k^{j'}}{j'!}
         \frac{e^{-(20-j)}(20-j)^{k'}}{k'!}

    i.e., the in->out correlation is "disassortative", the out->in correlation
    is "assortative", and everything else is uncorrelated.
    We will use a flat degree distribution in the range [1,20).

    >>> p = scipy.stats.poisson
    >>> g = gt.random_graph(20000, lambda: (sample_k(19), sample_k(19)),
    ...                                     lambda a,b: (p.pmf(a[0],b[1])*
    ...                                                  p.pmf(a[1],20-b[0])))

    Lets plot the average degree correlations to check.

    >>> clf()
    >>> corr = gt.avg_neighbour_corr(g, "in", "in")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...         label=r"$\left<\text{in}\right>$ vs in")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...         label=r"$\left<\text{out}\right>$ vs in")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{in}\right>$ vs out")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{out}\right>$ vs out")
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    (...)
    >>> legend(loc="best")
    <...>
    >>> xlabel("source degree")
    <...>
    >>> ylabel("average target degree")
    <...>
    >>> savefig("deg-corr-dir.png")

    .. figure:: deg-corr-dir.png
        :align: center

        Average nearest neighbour correlations.
    """
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    seed = numpy.random.randint(0, sys.maxint)
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    g = Graph()
    if deg_corr == None:
        uncorrelated = True
    else:
        uncorrelated = False
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    if not directed and deg_corr != None:
        corr = lambda i,j: _corr_wrap(i, j, deg_corr)
    else:
        corr = deg_corr
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    libgraph_tool_generation.gen_random_graph(g._Graph__graph, N,
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                                              deg_sampler, corr,
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                                              uncorrelated, not parallel,
                                              not self_loops, not directed,
                                              seed, verbose)
    g.set_directed(directed)
    return g
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def random_rewire(g, strat= "uncorrelated", parallel_edges = False,
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                  self_loops = False):
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    r"""
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    Shuffle the graph in-place. The degrees (either in or out) of each vertex
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    are always the same, but otherwise the edges are randomly placed. If
    strat == "correlated", the degree correlations are also maintained: The new
    source and target of each edge both have the same in and out-degree.

    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be shuffled. The graph will be modified.
    strat : string (optional, default: "uncorrelated")
        If strat == "uncorrelated" only the degrees of the vertices will be
        maintained, nothing else. If strat == "correlated", additionally the new
        source and target of each edge both have the same in and out-degree.
    parallel : bool (optional, default: False)
        If True, parallel edges are allowed.
    self_loops : bool (optional, default: False)
        If True, self-loops are allowed.

    See Also
    --------
    random_graph: random graph generation

    Notes
    -----

    Each edge gets swapped at least once, so the overall complexity is
    :math:`O(E)`.

    Examples
    --------

    Some small graphs for visualization.

    >>> from numpy.random import zipf, seed
    >>> from pylab import *
    >>> seed(42)
    >>> g = gt.random_graph(1000, lambda: sample_k(10),
    ...                     lambda i,j: exp(abs(i-j)), directed=False)
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    >>> gt.graph_draw(g, layout="arf", output="rewire_orig.png", size=(6,6))
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    <...>
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    >>> gt.random_rewire(g, "correlated")
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    >>> gt.graph_draw(g, layout="arf", output="rewire_corr.png", size=(6,6))
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    <...>
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    >>> gt.random_rewire(g)
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    >>> gt.graph_draw(g, layout="arf", output="rewire_uncorr.png", size=(6,6))
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    <...>
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    Some `ridiculograms <http://www.youtube.com/watch?v=YS-asmU3p_4>`_ :
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    .. image:: rewire_orig.png
    .. image:: rewire_corr.png
    .. image:: rewire_uncorr.png
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    *Left:* Original graph; *Middle:* Shuffled graph, with degree
    correlations; *Right:* Shuffled graph, without degree correlations.
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    We can try some larger graphs to get better statistics.

    >>> clf()
    >>> g = gt.random_graph(20000, lambda: sample_k(20),
    ...                     lambda i,j: exp(abs(i-j)), directed=False)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="original")
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    (...)
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="correlated")
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    (...)
    >>> gt.random_rewire(g)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="uncorrelated")
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    (...)
    >>> xlabel("$k$")
    <...>
    >>> ylabel(r"$\left<k_{nn}\right>$")
    <...>
    >>> legend(loc="best")
    <...>
    >>> savefig("shuffled-stats.png")

    .. figure:: shuffled-stats.png
        :align: center

        Average degree correlations for the different shuffled and non-shuffled
        graphs. The shuffled graph with correlations displays exactly the same
        correlation as the original graph.

    Now let's do it for a directed graph. See
    :func:`~graph_tool.generation.random_graph` for more details.

    >>> p = scipy.stats.poisson
    >>> g = gt.random_graph(20000, lambda: (sample_k(19), sample_k(19)),
    ...                                     lambda a,b: (p.pmf(a[0],b[1])*
    ...                                                  p.pmf(a[1],20-b[0])))
    >>> clf()
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{o}\right>$ vs i")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
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    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{i}\right>$ vs o")
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    (...)
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
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    ...          label=r"$\left<\text{o}\right>$ vs i, corr.")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
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    ...          label=r"$\left<\text{i}\right>$ vs o, corr.")
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    (...)
    >>> gt.random_rewire(g, "uncorrelated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
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    ...          label=r"$\left<\text{o}\right>$ vs i, uncorr.")
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    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
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    ...          label=r"$\left<\text{i}\right>$ vs o, uncorr.")
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    (...)
    >>> legend(loc="best")
    <...>
    >>> xlabel("source degree")
    <...>
    >>> ylabel("average target degree")
    <...>
    >>> savefig("shuffled-deg-corr-dir.png")

    .. figure:: shuffled-deg-corr-dir.png
        :align: center

        Average degree correlations for the different shuffled and non-shuffled
        directed graphs. The shuffled graph with correlations displays exactly
        the same correlation as the original graph.
    """

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    seed = numpy.random.randint(0, sys.maxint)
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    g.stash_filter(reversed=True)
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    libgraph_tool_generation.random_rewire(g._Graph__graph, strat, self_loops,
                                           parallel_edges, seed)
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    g.pop_filter(reversed=True)
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def predecessor_tree(g, pred_map):
    """Return a graph from a list of predecessors given by
    the 'pred_map' vertex property."""

    _check_prop_scalar(pred_map, "pred_map")
    pg = Graph()
    libgraph_tool_generation.predecessor_graph(g._Graph__graph,
                                               pg._Graph__graph,
                                               _prop("v", g, pred_map))
    return pg
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def line_graph(g):
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    """Return the line graph of the given graph `g`.

    Notes
    -----
    Given an undirected graph G, its line graph L(G) is a graph such that

        * each vertex of L(G) represents an edge of G; and
        * two vertices of L(G) are adjacent if and only if their corresponding
          edges share a common endpoint ("are adjacent") in G.

    For a directed graph, the second criterion becomes:

       * Two vertices representing directed edges from u to v and from w to x in
         G are connected by an edge from uv to wx in the line digraph when v =
         w.

    References
    ----------
    .. [line-wiki] http://en.wikipedia.org/wiki/Line_graph
    """
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    lg = Graph(directed=g.is_directed())

    vertex_map = lg.new_vertex_property("int64_t")

    libgraph_tool_generation.line_graph(g._Graph__graph,
                                        lg._Graph__graph,
                                        _prop("v", lg, vertex_map))
    return lg, vertex_map
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def graph_union(g1, g2, props=[], include=False):
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    """Return the union of graphs g1 and g2, composed of all edges and vertices
    of g1 and g2, without overlap.

    Parameters
    ----------
    g1 : :class:`~graph_tool.Graph`
       First graph in the union.
    g2 : :class:`~graph_tool.Graph`
       Second graph in the union.
    props : list of tuples of :class:`~graph_tool.PropertyMap` (optional, default: [])
       Each element in this list must be a tuple of two PropertyMap objects. The
       first element must be a property of `g1`, and the second of `g2`. The
       values of the property maps are propagated into the union graph, and
       returned.
    include : bool (optional, default: False)
       If true, graph `g2` is inserted into `g1` which is modified. If false, a
       new graph is created, and both graphs remain unmodified.

    Returns
    -------
    ug : :class:`~graph_tool.Graph`
        The union graph
    props : list of :class:`~graph_tool.PropertyMap` objects
        List of propagated properties.  This is only returned if `props` is not
        empty.
    """
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    if not include:
        g1 = Graph(g1)
    g1.stash_filter(directed=True)
    g1.set_directed(True)
    g2.stash_filter(directed=True)
    g2.set_directed(True)
    n_props = []

    try:
        vmap, emap = libgraph_tool_generation.graph_union(g1._Graph__graph,
                                                          g2._Graph__graph)
        for p in props:
            p1, p2 = p
            if not include:
                p1 = g1.copy_property(p1)
            if p2.value_type() != p1.value_type():
                p2 = g2.copy_property(p2, value_type=p1.value_type())
            if p1.key_type() == 'v':
                libgraph_tool_generation.\
                      vertex_property_union(g1._Graph__graph, g2._Graph__graph,
                                            vmap, emap,
                                            _prop(p1.key_type(), g1, p1),
                                            _prop(p2.key_type(), g2, p2))
            else:
                libgraph_tool_generation.\
                      edge_property_union(g1._Graph__graph, g2._Graph__graph,
                                          vmap, emap,
                                          _prop(p1.key_type(), g1, p1),
                                          _prop(p2.key_type(), g2, p2))
            n_props.append(p1)
    finally:
        g1.pop_filter(directed=True)
        g2.pop_filter(directed=True)

    if len(n_props) > 0:
        return g1, n_props
    else:
        return g1
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@_limit_args({"type":["simple", "delaunay"]})
def triangulation(points, type="simple"):
    r"""
    Generate a 2D or 3D triangulation graph from a given point set.

    Parameters
    ----------
    points : :class:`~numpy.ndarray`
        Point set for the triangulation. It may be either a N x d array, where N
        is the number of points, and d is the space dimension (either 2 or 3).
    type : string (optional, default: 'simple')
        Type of triangulation. May be either 'simple' or 'delaunay'.

    Returns
    -------
    triangulation_graph : :class:`~graph_tool.Graph`
        The generated graph.
    pos : :class:`~graph_tool.PropertyMap`
        Vertex property map with the Cartesian coordinates.

    See Also
    --------
    random_graph: random graph generation

    Notes
    -----

    A triangulation [cgal_triang]_ is division of the convex hull of a point set
    into triangles, using only that set as triangle vertices.

    In simple triangulations (`type="simple"`), the insertion of a point is done
    by locating a face that contains the point, and splitting this face into
    three new faces (the order of insertion is therefore important). If the
    point falls outside the convex hull, the triangulation is restored by
    flips. Apart from the location, insertion takes a time O(1). This bound is
    only an amortized bound for points located outside the convex hull.

    Delaunay triangulations (`type="delaunay"`) have the specific empty sphere
    property, that is, the circumscribing sphere of each cell of such a
    triangulation does not contain any other vertex of the triangulation in its
    interior. These triangulations are uniquely defined except in degenerate
    cases where five points are co-spherical. Note however that the CGAL
    implementation computes a unique triangulation even in these cases.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> points = random((500,2))
    >>> for i in xrange(points.shape[0]):
    ...     x,y = 1,1
    ...     while sqrt(x**2 + y**2) > 0.5:
    ...          x, y = random()-0.5, random()-0.5
    ...     points[i,:] = [x,y]
    >>> g, pos = gt.triangulation(points)
    >>> b = gt.betweenness(g)
    >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.08, vcolor=b[0],
    ...               ecolor=b[1], output="triang.png")
    <...>
    >>> g, pos = gt.triangulation(points, type="delaunay")
    >>> b = gt.betweenness(g)
    >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.08, vcolor=b[0],
    ...               ecolor=b[1], output="triang-delaunay.png")
    <...>

    2D triangulation of random points:

    .. image:: triang.png
    .. image:: triang-delaunay.png

    *Left:* Simple triangulation. *Right:* Delaunay triangulation. The colors
    correspond to the betweeness centrality.

    References
    ----------
    .. [cgal_triang] http://www.cgal.org/Manual/last/doc_html/cgal_manual/Triangulation_3/Chapter_main.html

    """

    if points.shape[1] not in [2,3]:
        raise ValueError("points array must have shape N x d, with d either 2 or 3.")
    # copy points to ensure continuity and correct data type
    points = numpy.array(points, dtype='float64')
    if points.shape[1] == 2:
        npoints = numpy.zeros((points.shape[0], 3))
        npoints[:,:2] = points
        points = npoints
    g = Graph(directed=False)
    pos = g.new_vertex_property("vector<double>")
    libgraph_tool_generation.triangulation(g._Graph__graph, points,
                                           _prop("v", g, pos), type)
    return g, pos