#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2007-2011 Tiago de Paula Peixoto
#
# 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 .
"""
``graph_tool.generation`` - Random graph generation
---------------------------------------------------
Summary
+++++++
.. autosummary::
:nosignatures:
random_graph
random_rewire
predecessor_tree
line_graph
graph_union
triangulation
lattice
geometric_graph
price_network
Contents
++++++++
"""
from .. dl_import import dl_import
dl_import("import libgraph_tool_generation")
from .. import Graph, GraphView, _check_prop_scalar, _prop, _limit_args, _gt_type
from .. stats import label_parallel_edges, label_self_loops
import inspect
import types
import sys, numpy, numpy.random
__all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph",
"graph_union", "triangulation", "lattice", "geometric_graph",
"price_network"]
def random_graph(N, deg_sampler, deg_corr=None, directed=True,
parallel_edges=False, self_loops=False, blockmodel=None,
random=True, mix_time=10, verbose=False):
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.
Optionally, you can also pass a function which receives one argument. In
this case the argument passed will be the index of the vertex which will
receive the degree. If ``blockmodel != None``, the value passed will be
the block value of the respective vertex instead.
deg_corr : function (optional, default: ``None``)
A function which gives 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.
If ``blockmodel != None``, the value passed to the function will be the
block value of the respective vertices, not the in/out-degree pairs.
directed : bool (optional, default: ``True``)
Whether the generated graph should be directed.
parallel_edges : bool (optional, default: ``False``)
If ``True``, parallel edges are allowed.
self_loops : bool (optional, default: ``False``)
If ``True``, self-loops are allowed.
blockmodel : list or :class:`~numpy.ndarray` or function (optional, default: ``None``)
If supplied, the graph will be sampled from a blockmodel ensemble. If
the value is a list or a :class:`~numpy.ndarray`, it must have
``len(block_model) == N``, and the values will define to which block
each vertex belongs.
If this value is a function, it will be used to sample the block
types. It must be callable either with no arguments or with a single
argument which will be the vertex index. In either case it must return
an integer.
random : bool (optional, default: ``True``)
If ``True``, the returned graph is randomized. Otherwise a deterministic
placement of the edges will be used.
mix_time : int (optional, default: ``10``)
Number of edge sweeps to perform in order to mix the graph. This value
is ignored if ``parallel_edges == self_loops == True`` and
``strat != "probabilistic"``.
verbose : bool (optional, default: ``False``)
If ``True``, verbose information is displayed.
Returns
-------
random_graph : :class:`~graph_tool.Graph`
The generated graph.
blocks : :class:`~graph_tool.PropertyMap`
A vertex property map with the block values. This is only returned if
``blockmodel != None``.
See Also
--------
random_rewire: in place graph shuffling
Notes
-----
The algorithm makes sure the degree sequence is graphical (i.e. realizable)
and keeps re-sampling the degrees if is not. With a valid degree sequence,
the edges are placed deterministically, and later the graph is shuffled with
the :func:`~graph_tool.generation.random_rewire` function, with the
``mix_time`` parameter passed as ``n_iter``.
The complexity is :math:`O(V + E)` if parallel edges are allowed, and
:math:`O(V + E \times\text{mix-time})` if parallel edges are not allowed.
.. note ::
If ``parallel_edges == False`` this algorithm only guarantees that the
returned graph will be a random sample from the desired ensemble if
``mix_time`` is sufficiently large. The algorithm implements an
efficient Markov chain based on edge swaps, with a mixing time which
depends on the degree distribution and correlations desired. If degree
correlations are provided, the mixing time tends to be larger.
If ``strat == "probabilistic"``, the Markov chain still needs to be
mixed, even if parallel edges and self-loops are allowed. In this case
the Markov chain is implemented using the Metropolis-Hastings
[metropolis-equations-1953]_ [hastings-monte-carlo-1970]_
acceptance/rejection algorithm.
Examples
--------
>>> from numpy.random import randint, random, seed, poisson
>>> from pylab import *
>>> seed(43)
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,
... mix_time=100)
>>> gt.scalar_assortativity(g, "out")
(0.6435658697163692, 0.010420519538259333)
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()
<...>
>>> xlabel("in-degree")
<...>
>>> ylabel("out-degree")
<...>
>>> savefig("combined-deg-hist.pdf")
.. figure:: combined-deg-hist.*
: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])),
... mix_time=100)
Lets plot the average degree correlations to check.
>>> figure(figsize=(6,3))
<...>
>>> axes([0.1,0.15,0.63,0.8])
<...>
>>> corr = gt.avg_neighbour_corr(g, "in", "in")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{in}\right>$ vs in")
(...)
>>> corr = gt.avg_neighbour_corr(g, "in", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{out}\right>$ vs in")
(...)
>>> corr = gt.avg_neighbour_corr(g, "out", "in")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{in}\right>$ vs out")
(...)
>>> corr = gt.avg_neighbour_corr(g, "out", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{out}\right>$ vs out")
(...)
>>> legend(loc=(1.05,0.5))
<...>
>>> xlabel("source degree")
<...>
>>> ylabel("average target degree")
<...>
>>> savefig("deg-corr-dir.pdf")
.. figure:: deg-corr-dir.*
:align: center
Average nearest neighbour correlations.
**Blockmodels**
The following example shows how a blockmodel [#]_ can be generated. We will
consider a system of 10 blocks, which form communities. The connection
probability will be given by
>>> def corr(a, b):
... if a == b:
... return 0.999
... else:
... return 0.001
The blockmodel can be generated as follows.
>>> g, bm = gt.random_graph(1000, lambda: poisson(10), directed=False,
... blockmodel=lambda: randint(10), deg_corr=corr,
... mix_time=500)
>>> gt.graph_draw(g, vcolor=bm, layout="sfdp", output="blockmodel.pdf")
<...>
.. figure:: blockmodel.*
:align: center
Simple blockmodel with 10 blocks.
.. [#] Blockmodels are equivalent to the hidden variable model
[boguna-correlated-2003]_.
References
----------
.. [metropolis-equations-1953] Metropolis, N.; Rosenbluth, A.W.;
Rosenbluth, M.N.; Teller, A.H.; Teller, E. "Equations of State
Calculations by Fast Computing Machines". Journal of Chemical Physics 21
(6): 1087–1092 (1953). :doi:`10.1063/1.1699114`
.. [hastings-monte-carlo-1970] Hastings, W.K. "Monte Carlo Sampling Methods
Using Markov Chains and Their Applications". Biometrika 57 (1): 97–109 (1970).
:doi:`10.1093/biomet/57.1.97`
.. [boguna-correlated-2003] M. Boguñá and R. Pastor-Satorras, "Class of
correlated random networks with hidden variables" Physical Review E
68, 036112 (2003) :doi:`10.1103/PhysRevE.68.036112`
"""
seed = numpy.random.randint(0, sys.maxint)
g = Graph()
if deg_corr == None:
uncorrelated = True
else:
uncorrelated = False
if (type(blockmodel) is types.FunctionType or
type(blockmodel) is types.LambdaType):
bm = numpy.zeros(N, dtype="int")
if len(inspect.getargspec(blockmodel)[0]) == 0:
for i in xrange(N):
bm[i] = blockmodel()
else:
for i in xrange(N):
bm[i] = blockmodel(i)
blockmodel = numpy.array(bm)
if len(inspect.getargspec(deg_sampler)[0]) > 0:
if blockmodel is not None:
sampler = lambda i: deg_sampler(blockmodel[i])
else:
sampler = deg_sampler
else:
sampler = lambda i: deg_sampler()
libgraph_tool_generation.gen_graph(g._Graph__graph, N, sampler,
uncorrelated, not parallel_edges,
not self_loops, not directed,
seed, verbose, True)
g.set_directed(directed)
if blockmodel is not None:
btype = _gt_type(blockmodel[0])
bm = g.new_vertex_property(btype)
if btype in ["object", "string"] or "vector" in btype:
for v in g.vertices():
bm[v] = blockmodel[int(v)]
else:
try:
bm.a = blockmodel
except ValueError:
bm = g.new_vertex_property("object")
for v in g.vertices():
bm[v] = blockmodel[int(v)]
else:
bm = None
if parallel_edges and self_loops and deg_corr is None:
mix_time = 1
if random:
if deg_corr is not None:
random_rewire(g, strat="probabilistic", n_iter=mix_time,
parallel_edges=parallel_edges, deg_corr=deg_corr,
self_loops=self_loops, blockmodel=bm,
verbose=verbose)
else:
random_rewire(g, parallel_edges=parallel_edges, n_iter=mix_time,
self_loops=self_loops, verbose=verbose)
if bm is None:
return g
else:
return g, bm
@_limit_args({"strat": ["erdos", "correlated", "uncorrelated", "probabilistic"]})
def random_rewire(g, strat="uncorrelated", n_iter=1, edge_sweep=True,
parallel_edges=False, self_loops=False, deg_corr=None,
blockmodel=None, ret_fail=False, verbose=False):
r"""
Shuffle the graph in-place.
If ``strat != "erdos"``, the degrees (either in or out) of each vertex 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. If
``strat == "probabilistic"``, then edges are rewired according to the degree
correlation given by the parameter ``deg_corr``.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be shuffled. The graph will be modified.
strat : string (optional, default: ``"uncorrelated"``)
If ``strat == "erdos"``, the resulting graph will be entirely random. 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.
If ``strat == "probabilistic"``, than edges are rewired according to the
degree correlation given by the parameter ``deg_corr``.
n_iter : int (optional, default: ``1``)
Number of iterations. If ``edge_sweep == True``, each iteration
corresponds to an entire "sweep" over all edges. Otherwise this
corresponds to the total number of edges which are randomly chosen for a
swap attempt (which may repeat).
edge_sweep : bool (optional, default: ``True``)
If ``True``, each iteration will perform an entire "sweep" over the
edges, where each edge is visited once in random order, and a edge swap
is attempted.
parallel : bool (optional, default: ``False``)
If ``True``, parallel edges are allowed.
self_loops : bool (optional, default: ``False``)
If ``True``, self-loops are allowed.
deg_corr : function (optional, default: ``None``)
A function which gives 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. This parameter is ignored,
unless ``strat == "probabilistic"``.
If ``blockmodel != None``, the value passed to the function will be the
block value of the respective vertices, not the in/out-degree pairs.
blockmodel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
If supplied, the graph will be rewired to conform to a blockmodel
ensemble. The value must be a vertex property map which defines the
block of each vertex.
ret_fail : bool (optional, default: ``False``)
If ``True``, the number of failed edge moves (due to parallel edges or
self-loops) is returned.
verbose : bool (optional, default: ``False``)
If ``True``, verbose information is displayed.
Returns
-------
fail_count : int
Number of failed edge moves (due to parallel edges or self-loops).
See Also
--------
random_graph: random graph generation
Notes
-----
This algorithm iterates through all the edges in the network and tries to
swap its target or source with the target or source of another edge.
.. note::
If ``parallel_edges = False``, parallel edges are not placed during
rewiring. In this case, the returned graph will be a uncorrelated sample
from the desired ensemble only if ``n_iter`` is sufficiently large. The
algorithm implements an efficient Markov chain based on edge swaps, with
a mixing time which depends on the degree distribution and correlations
desired. If degree probabilistic correlations are provided, the mixing
time tends to be larger.
If ``strat == "probabilistic"``, the Markov chain still needs to be
mixed, even if parallel edges and self-loops are allowed. In this case
the Markov chain is implemented using the Metropolis-Hastings
[metropolis-equations-1953]_ [hastings-monte-carlo-1970]_
acceptance/rejection algorithm.
Each edge is tentatively swapped once per iteration, so the overall
complexity is :math:`O(V + E \times \text{n-iter})`. If ``edge_sweep ==
False``, the complexity becomes :math:`O(V + E + \text{n-iter})`.
Examples
--------
Some small graphs for visualization.
>>> from numpy.random import random, seed
>>> from pylab import *
>>> seed(43)
>>> g, pos = gt.triangulation(random((1000,2)))
>>> gt.graph_draw(g, layout="arf", output="rewire_orig.pdf", size=(6,6))
<...>
>>> gt.random_rewire(g, "correlated")
>>> gt.graph_draw(g, layout="arf", output="rewire_corr.pdf", size=(6,6))
<...>
>>> gt.random_rewire(g)
>>> gt.graph_draw(g, layout="arf", output="rewire_uncorr.pdf", size=(6,6))
<...>
>>> gt.random_rewire(g, "erdos")
>>> gt.graph_draw(g, layout="arf", output="rewire_erdos.pdf", size=(6,6))
<...>
Some `ridiculograms `_ :
.. image:: rewire_orig.*
.. image:: rewire_corr.*
.. image:: rewire_uncorr.*
.. image:: rewire_erdos.*
*From left to right:* Original graph --- Shuffled graph, with degree
correlations --- Shuffled graph, without degree correlations --- Shuffled graph,
with random degrees.
We can try some larger graphs to get better statistics.
>>> figure()
<...>
>>> g = gt.random_graph(30000, lambda: sample_k(20),
... lambda i, j: exp(abs(i-j)), directed=False,
... mix_time=100)
>>> corr = gt.avg_neighbour_corr(g, "out", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="original")
(...)
>>> gt.random_rewire(g, "correlated")
>>> corr = gt.avg_neighbour_corr(g, "out", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="*", label="correlated")
(...)
>>> gt.random_rewire(g)
>>> corr = gt.avg_neighbour_corr(g, "out", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="uncorrelated")
(...)
>>> gt.random_rewire(g, "erdos")
>>> corr = gt.avg_neighbour_corr(g, "out", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="Erdos")
(...)
>>> xlabel("$k$")
<...>
>>> ylabel(r"$\left$")
<...>
>>> legend(loc="best")
<...>
>>> savefig("shuffled-stats.pdf")
.. figure:: shuffled-stats.*
: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])),
... mix_time=100)
>>> figure(figsize=(6,3))
<...>
>>> axes([0.1,0.15,0.6,0.8])
<...>
>>> corr = gt.avg_neighbour_corr(g, "in", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{o}\right>$ vs i")
(...)
>>> corr = gt.avg_neighbour_corr(g, "out", "in")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{i}\right>$ vs o")
(...)
>>> gt.random_rewire(g, "correlated")
>>> corr = gt.avg_neighbour_corr(g, "in", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{o}\right>$ vs i, corr.")
(...)
>>> corr = gt.avg_neighbour_corr(g, "out", "in")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{i}\right>$ vs o, corr.")
(...)
>>> gt.random_rewire(g, "uncorrelated")
>>> corr = gt.avg_neighbour_corr(g, "in", "out")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{o}\right>$ vs i, uncorr.")
(...)
>>> corr = gt.avg_neighbour_corr(g, "out", "in")
>>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
... label=r"$\left<\text{i}\right>$ vs o, uncorr.")
(...)
>>> legend(loc=(1.05,0.45))
<...>
>>> xlabel("source degree")
<...>
>>> ylabel("average target degree")
<...>
>>> savefig("shuffled-deg-corr-dir.pdf")
.. figure:: shuffled-deg-corr-dir.*
: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.
References
----------
.. [metropolis-equations-1953] Metropolis, N.; Rosenbluth, A.W.;
Rosenbluth, M.N.; Teller, A.H.; Teller, E. "Equations of State
Calculations by Fast Computing Machines". Journal of Chemical Physics 21
(6): 1087–1092 (1953). :doi:`10.1063/1.1699114`
.. [hastings-monte-carlo-1970] Hastings, W.K. "Monte Carlo Sampling Methods
Using Markov Chains and Their Applications". Biometrika 57 (1): 97–109 (1970).
:doi:`10.1093/biomet/57.1.97`
"""
seed = numpy.random.randint(0, sys.maxint)
if not parallel_edges:
p = label_parallel_edges(g)
if p.a.max() != 0:
raise ValueError("Parallel edge detected. Can't rewire " +
"graph without parallel edges if it " +
"already contains parallel edges!")
if not self_loops:
l = label_self_loops(g)
if l.a.max() != 0:
raise ValueError("Self-loop detected. Can't rewire graph " +
"without self-loops if it already contains" +
" self-loops!")
if (deg_corr is not None and not g.is_directed()) and blockmodel is None:
corr = lambda i, j: deg_corr(i[1], j[1])
else:
corr = deg_corr
if strat != "probabilistic":
g = GraphView(g, reversed=False)
elif blockmodel is not None:
strat = "blockmodel"
pcount = libgraph_tool_generation.random_rewire(g._Graph__graph, strat,
n_iter, not edge_sweep,
self_loops, parallel_edges,
corr, _prop("v", g, blockmodel),
seed, verbose)
if ret_fail:
return pcount
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
def line_graph(g):
"""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
"""
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
def graph_union(g1, g2, props=None, include=False):
"""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.
Examples
--------
>>> from numpy.random import random, seed
>>> seed(42)
>>> g = gt.triangulation(random((300,2)))[0]
>>> ug = gt.graph_union(g, g)
>>> uug = gt.graph_union(g, ug)
>>> gt.graph_draw(g, layout="arf", size=(8,8), output="graph_original.pdf")
<...>
>>> gt.graph_draw(ug, layout="arf", size=(8,8), output="graph_union.pdf")
<...>
>>> gt.graph_draw(uug, layout="arf", size=(8,8), output="graph_union2.pdf")
<...>
.. image:: graph_original.*
.. image:: graph_union.*
.. image:: graph_union2.*
"""
if props == None:
props = []
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
@_limit_args({"type": ["simple", "delaunay"]})
def triangulation(points, type="simple", periodic=False):
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'.
periodic : bool (optional, default: ``False``)
If ``True``, periodic boundary conditions will be used. This is
parameter is valid only for type="delaunay", and is otherwise ignored.
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 a 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)) * 4
>>> g, pos = gt.triangulation(points)
>>> weight = g.new_edge_property("double") # Edge weights corresponding to
... # Euclidean distances
>>> for e in g.edges():
... weight[e] = sqrt(sum((array(pos[e.source()]) -
... array(pos[e.target()]))**2))
>>> b = gt.betweenness(g, weight=weight)
>>> b[1].a *= 100
>>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0],
... eprops={"penwidth":b[1]}, output="triang.pdf")
<...>
>>> g, pos = gt.triangulation(points, type="delaunay")
>>> weight = g.new_edge_property("double")
>>> for e in g.edges():
... weight[e] = sqrt(sum((array(pos[e.source()]) -
... array(pos[e.target()]))**2))
>>> b = gt.betweenness(g, weight=weight)
>>> b[1].a *= 120
>>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0],
... eprops={"penwidth":b[1]}, output="triang-delaunay.pdf")
<...>
2D triangulation of random points:
.. image:: triang.*
.. image:: triang-delaunay.*
*Left:* Simple triangulation. *Right:* Delaunay triangulation. The vertex
colors and the edge thickness correspond to the weighted betweenness
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")
libgraph_tool_generation.triangulation(g._Graph__graph, points,
_prop("v", g, pos), type, periodic)
return g, pos
def lattice(shape, periodic=False):
r"""
Generate a N-dimensional square lattice.
Parameters
----------
shape : list or :class:`~numpy.ndarray`
List of sizes in each dimension.
periodic : bool (optional, default: ``False``)
If ``True``, periodic boundary conditions will be used.
Returns
-------
lattice_graph : :class:`~graph_tool.Graph`
The generated graph.
See Also
--------
triangulation: 2D or 3D triangulation
random_graph: random graph generation
Examples
--------
>>> g = gt.lattice([10,10])
>>> gt.graph_draw(g, size=(8,8), output="lattice.pdf")
<...>
>>> g = gt.lattice([10,20], periodic=True)
>>> gt.graph_draw(g, size=(8,8), output="lattice_periodic.pdf")
<...>
>>> g = gt.lattice([10,10,10])
>>> gt.graph_draw(g, size=(8,8), output="lattice_3d.pdf")
<...>
.. image:: lattice.*
.. image:: lattice_periodic.*
.. image:: lattice_3d.*
*Left:* 10x10 2D lattice. *Middle:* 10x20 2D periodic lattice (torus).
*Right:* 10x10x10 3D lattice.
References
----------
.. [lattice] http://en.wikipedia.org/wiki/Square_lattice
"""
g = Graph(directed=False)
libgraph_tool_generation.lattice(g._Graph__graph, shape, periodic)
return g
def geometric_graph(points, radius, ranges=None):
r"""
Generate a geometric network form a set of N-dimensional points.
Parameters
----------
points : list or :class:`~numpy.ndarray`
List of points. This must be a two-dimensional array, where the rows are
coordinates in a N-dimensional space.
radius : float
Pairs of points with an euclidean distance lower than this parameters
will be connected.
ranges : list or :class:`~numpy.ndarray` (optional, default: ``None``)
If provided, periodic boundary conditions will be assumed, and the
values of this parameter it will be used as the ranges in all
dimensions. It must be a two-dimensional array, where each row will
cointain the lower and upper bound of each dimension.
Returns
-------
geometric_graph : :class:`~graph_tool.Graph`
The generated graph.
pos : :class:`~graph_tool.PropertyMap`
A vertex property map with the position of each vertex.
Notes
-----
A geometric graph [geometric-graph]_ is generated by connecting points
embedded in a N-dimensional euclidean space which are at a distance equal to
or smaller than a given radius.
See Also
--------
triangulation: 2D or 3D triangulation
random_graph: random graph generation
lattice : N-dimensional square lattice
Examples
--------
>>> from numpy.random import seed, random
>>> seed(42)
>>> points = random((500, 2)) * 4
>>> g, pos = gt.geometric_graph(points, 0.3)
>>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), output="geometric.pdf")
<...>
>>> g, pos = gt.geometric_graph(points, 0.3, [(0,4), (0,4)])
>>> gt.graph_draw(g, size=(8,8), output="geometric_periodic.pdf")
<...>
.. image:: geometric.*
.. image:: geometric_periodic.*
*Left:* Geometric network with random points. *Right:* Same network, but
with periodic boundary conditions.
References
----------
.. [geometric-graph] Jesper Dall and Michael Christensen, "Random geometric
graphs", Phys. Rev. E 66, 016121 (2002), :doi:`10.1103/PhysRevE.66.016121`
"""
g = Graph(directed=False)
pos = g.new_vertex_property("vector")
if type(points) != numpy.ndarray:
points = numpy.array(points)
if len(points.shape) < 2:
raise ValueError("points list must be a two-dimensional array!")
if ranges is not None:
periodic = True
if type(ranges) != numpy.ndarray:
ranges = numpy.array(ranges, dtype="float")
else:
ranges = array(ranges, dtype="float")
else:
periodic = False
ranges = ()
libgraph_tool_generation.geometric(g._Graph__graph, points, float(radius),
ranges, periodic,
_prop("v", g, pos))
return g, pos
def price_network(N, m=1, c=None, gamma=1, directed=True, seed_graph=None):
r"""A generalized version of Price's -- or Barabási-Albert if undirected -- preferential attachment network model.
Parameters
----------
N : int
Size of the network.
m : int (optional, default: ``1``)
Out-degree of newly added vertices.
c : float (optional, default: ``1 if directed == True else 0``)
Constant factor added to the probability of a vertex receiving an edge
(see notes below).
gamma : float (optional, default: ``1``)
Preferential attachment power (see notes below).
directed : bool (optional, default: ``True``)
If ``True``, a Price network is generated. If ``False``, a
Barabási-Albert network is generated.
seed_graph : :class:`~graph_tool.Graph` (optional, default: ``None``)
If provided, this graph will be used as the starting point of the
algorithm.
Returns
-------
price_graph : :class:`~graph_tool.Graph`
The generated graph.
Notes
-----
The (generalized) [price]_ network is either a directed or undirected graph
(the latter is called a Barabási-Albert network), generated dynamically by
at each step adding a new vertex, and connecting it to :math:`m` other
vertices, chosen with probability :math:`\pi` defined as:
.. math::
\pi \propto k^\gamma + c
where :math:`k` is the in-degree of the vertex (or simply the degree in the
undirected case). If :math:`\gamma=1`, the tail of resulting in-degree
distribution of the directed case is given by
.. math::
P_{k_\text{in}} \sim k_\text{in}^{-(2 + c/m)},
or for the undirected case
.. math::
P_{k} \sim k^{-(3 + c/m)}.
However, if :math:`\gamma \ne 1`, the in-degree distribution is not
scale-free (see [dorogovtsev-evolution]_ for details).
Note that if `seed_graph` is not given, the algorithm will *always* start
with one node if :math:`c > 0`, or with two nodes with a link between them
otherwise. If :math:`m > 1`, the degree of the newly added vertices will be
vary dynamically as :math:`m'(t) = \min(m, N(t))`, where :math:`N(t)` is the
number of vertices added so far. If this behaviour is undesired, a proper
seed graph with :math:`N \ge m` vertices must be provided.
This algorithm runs in :math:`O(N\log N)` time.
See Also
--------
triangulation: 2D or 3D triangulation
random_graph: random graph generation
lattice : N-dimensional square lattice
geometric_graph : N-dimensional geometric network
Examples
--------
>>> from numpy.random import seed, random
>>> seed(42)
>>> g = gt.price_network(100000)
>>> gt.graph_draw(g, layout="sfdp", size=(12,12), vcolor=g.vertex_index,
... output="price-network.pdf")
<...>
>>> g = gt.price_network(100000, c=0.1)
>>> gt.graph_draw(g, layout="sfdp", size=(12,12), vcolor=g.vertex_index,
... output="price-network-broader.pdf")
<...>
.. image:: price-network.*
.. image:: price-network-broader.*
Price networks with :math:`N=10^5` nodes. **Left:** :math:`c=1`, **Right:**
:math:`c=0.1`. The colors represent the order in which vertices were
added.
References
----------
.. [yule] Yule, G. U. "A Mathematical Theory of Evolution, based on the
Conclusions of Dr. J. C. Willis, F.R.S.". Philosophical Transactions of
the Royal Society of London, Ser. B 213: 21–87, 1925,
:doi:`10.1098/rstb.1925.0002`
.. [price] Derek De Solla Price, "A general theory of bibliometric and other
cumulative advantage processes", Journal of the American Society for
Information Science, Volume 27, Issue 5, pages 292–306, September 1976,
:doi:`10.1002/asi.4630270505`
.. [barabasi-albert] Barabási, A.-L., and Albert, R., "Emergence of
scaling in random networks", Science, 286, 509, 1999,
:doi:`10.1126/science.286.5439.509`
.. [dorogovtsev-evolution] S. N. Dorogovtsev and J. F. F. Mendes, "Evolution
of networks", Advances in Physics, 2002, Vol. 51, No. 4, 1079-1187,
:doi:`10.1080/00018730110112519`
"""
if c is None:
c = 1 if directed else 0
if seed_graph is None:
g = Graph(directed=directed)
if c > 0:
g.add_vertex()
else:
g.add_vertex(2)
g.add_edge(g.vertex(1), g.vertex(0))
N -= g.num_vertices()
else:
g = seed_graph
seed = numpy.random.randint(0, sys.maxint)
libgraph_tool_generation.price(g._Graph__graph, N, gamma, c, m, seed)
return g