#! /usr/bin/env python # -*- coding: utf-8 -*- # # graph_tool -- a general graph manipulation python module # # Copyright (C) 2007-2012 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, cache_probs=True, directed=True, parallel_edges=False, self_loops=False, blockmodel=None, block_type="int", degree_block=False, 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 or two arguments: If ``blockmodel == None``, the single argument passed will be the index of the vertex which will receive the degree. If ``blockmodel != None``, the first value passed will be the vertex index, and the second will be the block value of the vertex. 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. cache_probs : bool (optional, default: ``True``) If ``True``, the probabilities returned by the ``deg_corr`` parameter will be cached internally. This is crucial for good performance, since in this case the supplied python function is called only a few times, and not at every attempted edge rewire move. However, in the case were the different parameter combinations to the probability function is very large, the memory requirements to keep the cache may be very large. 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 a type compatible with the ``block_type`` parameter. block_type : string (optional, default: ``"int"``) Value type of block labels. Valid only if ``blockmodel != None``. degree_block : bool (optional, default: ``False``) If ``True``, the degree of each vertex will be appended to block labels when constructing the blockmodel, such that the resulting block type will be a pair :math:`(r, k)`, where :math:`r` is the original block label. 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") >>> >>> clf() >>> 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. >>> clf() >>> 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(bbox_to_anchor=(1.01, 0.5), loc="center left", borderaxespad=0.) <...> >>> 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 stochastic blockmodel [holland-stochastic-1983]_ [karrer-stochastic-2011]_ 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(5000, lambda: poisson(10), directed=False, ... blockmodel=lambda: randint(10), deg_corr=corr, ... mix_time=500) >>> gt.graph_draw(g, vertex_fill_color=bm, output="blockmodel.pdf") <...> .. figure:: blockmodel.* :align: center Simple blockmodel with 10 blocks. 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` .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt, "Stochastic blockmodels: First steps," Social Networks 5, no. 2: 109-13 (1983) :doi:`10.1016/0378-8733(83)90021-7` .. [karrer-stochastic-2011] Brian Karrer and M. E. J. Newman, "Stochastic blockmodels and community structure in networks," Physical Review E 83, no. 1: 016107 (2011) :doi:`10.1103/PhysRevE.83.016107` :arxiv:`1008.3926` """ 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): btype = block_type bm = [] if len(inspect.getargspec(blockmodel)[0]) == 0: for i in xrange(N): bm.append(blockmodel()) else: for i in xrange(N): bm.append(blockmodel(i)) blockmodel = bm elif blockmodel is not None: btype = _gt_type(blockmodel[0]) if len(inspect.getargspec(deg_sampler)[0]) > 0: if blockmodel is not None: sampler = lambda i: deg_sampler(i, 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 degree_block: if btype in ["object", "string"] or "vector" in btype: btype = "object" elif btype in ["int", "int32_t", "bool"]: btype = "vector" elif btype in ["long", "int64_t"]: btype = "vector" elif btype in ["double"]: btype = "vector" elif btype in ["long double"]: btype = "vector" if blockmodel is not None: bm = g.new_vertex_property(btype) if btype in ["object", "string"] or "vector" in btype: for v in g.vertices(): if not degree_block: bm[v] = blockmodel[int(v)] else: if g.is_directed(): bm[v] = (blockmodel[int(v)], v.in_degree(), v.out_degree()) else: bm[v] = (blockmodel[int(v)], v.out_degree()) 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, cache_probs=cache_probs, 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, cache_probs=True, 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. cache_probs : bool (optional, default: ``True``) If ``True``, the probabilities returned by the ``deg_corr`` parameter will be cached internally. This is crucial for good performance, since in this case the supplied python function is called only a few times, and not at every attempted edge rewire move. However, in the case were the different parameter combinations to the probability function is very large, the memory requirements to keep the cache may be very large. 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))) >>> pos = gt.arf_layout(g) >>> gt.graph_draw(g, pos=pos, output="rewire_orig.pdf", output_size=(200, 200)) <...> >>> gt.random_rewire(g, "correlated") >>> pos = gt.arf_layout(g) >>> gt.graph_draw(g, pos=pos, output="rewire_corr.pdf", output_size=(200, 200)) <...> >>> gt.random_rewire(g) >>> pos = gt.arf_layout(g) >>> gt.graph_draw(g, pos=pos, output="rewire_uncorr.pdf", output_size=(200, 200)) <...> >>> gt.random_rewire(g, "erdos") >>> pos = gt.arf_layout(g) >>> gt.graph_draw(g, pos=pos, output="rewire_erdos.pdf", output_size=(200, 200)) <...> 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 with larger graphs to get better statistics, as follows. >>> 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=r"Erd\H{o}s") <...> >>> 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() <...> >>> 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(bbox_to_anchor=(1.01, 0.5), loc="center left", borderaxespad=0.) <...> >>> 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` .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt, "Stochastic blockmodels: First steps," Social Networks 5, no. 2: 109-13 (1983) :doi:`10.1016/0378-8733(83)90021-7` .. [karrer-stochastic-2011] Brian Karrer and M. E. J. Newman, "Stochastic blockmodels and community structure in networks," Physical Review E 83, no. 1: 016107 (2011) :doi:`10.1103/PhysRevE.83.016107` :arxiv:`1008.3926` """ 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), cache_probs, 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, intersection=None, 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. intersection : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex property map owned by `g1` which maps each of each of its vertices to vertex indexes belonging to `g2`. Negative values mean no mapping exists, and thus both vertices in `g1` and `g2` will be present in the union graph. 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) >>> pos = gt.arf_layout(g) >>> gt.graph_draw(g, pos=pos, output_size=(300,300), output="graph_original.pdf") <...> >>> pos = gt.arf_layout(ug) >>> gt.graph_draw(ug, pos=pos, output_size=(300,300), output="graph_union.pdf") <...> >>> pos = gt.arf_layout(uug) >>> gt.graph_draw(uug, pos=pos, output_size=(300,300), output="graph_union2.pdf") <...> .. image:: graph_original.* .. image:: graph_union.* .. image:: graph_union2.* """ if props == None: props = [] if not include: g1 = Graph(g1) if intersection is None: intersection = g1.new_vertex_property("int32_t") intersection.a = 0 else: intersection = intersection.copy("int32_t") intersection.a[intersection.a >= 0] += 1 intersection.a[intersection.a < 0] = 0 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, _prop("v", g1, intersection)) for p1, p2 in props: 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, output_size=(300,300), vertex_fill_color=b[0], ... edge_pen_width=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, output_size=(300,300), vertex_fill_color=b[0], ... edge_pen_width=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, pos=gt.sfdp_layout(g, cooling_step=0.95, epsilon=1e-2), ... output_size=(300,300), output="lattice.pdf") <...> >>> g = gt.lattice([10,20], periodic=True) >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, cooling_step=0.95, epsilon=1e-2, multilevel=True), ... output_size=(300,300), output="lattice_periodic.pdf") <...> >>> g = gt.lattice([10,10,10]) >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, cooling_step=0.95, epsilon=1e-2, multilevel=True), ... output_size=(300,300), 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, output_size=(300,300), output="geometric.pdf") <...> >>> g, pos = gt.geometric_graph(points, 0.3, [(0,4), (0,4)]) >>> gt.graph_draw(g, output_size=(300,300), 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 -------- >>> g = gt.price_network(100000) >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, epsilon=1e-2, cooling_step=0.95), ... vertex_fill_color=g.vertex_index, vertex_size=2, ... edge_pen_width=1, output="price-network.png") <...> >>> g = gt.price_network(100000, c=0.1) >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, epsilon=1e-2, cooling_step=0.95), ... vertex_fill_color=g.vertex_index, vertex_size=2, ... edge_pen_width=1, output="price-network-broader.png") <...> .. figure:: price-network.png :align: center Price network with :math:`N=10^5` nodes and :math:`c=1`. The colors represent the order in which vertices were added. .. figure:: price-network-broader.png :align: center Price network with :math:`N=10^5` nodes and :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