Source code for graph_tool.topology

#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2006-2017 Tiago de Paula Peixoto <tiago@skewed.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/>.

"""
``graph_tool.topology`` - Assessing graph topology
--------------------------------------------------

Summary
+++++++

.. autosummary::
   :nosignatures:

   shortest_distance
   shortest_path
   all_shortest_paths
   all_predecessors
   all_paths
   all_circuits
   pseudo_diameter
   similarity
   vertex_similarity
   isomorphism
   subgraph_isomorphism
   mark_subgraph
   max_cardinality_matching
   max_independent_vertex_set
   min_spanning_tree
   random_spanning_tree
   dominator_tree
   topological_sort
   transitive_closure
   tsp_tour
   sequential_vertex_coloring
   label_components
   label_biconnected_components
   label_largest_component
   label_out_component
   vertex_percolation
   edge_percolation
   kcore_decomposition
   is_bipartite
   is_DAG
   is_planar
   make_maximal_planar
   edge_reciprocity

Contents
++++++++

"""

from __future__ import division, absolute_import, print_function

from .. dl_import import dl_import
dl_import("from . import libgraph_tool_topology")

from .. import _prop, Vector_int32_t, _check_prop_writable, \
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView,\
     libcore, _get_rng, _degree, perfect_prop_hash, _limit_args
from .. stats import label_self_loops
import random, sys, numpy, collections

__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
           "max_cardinality_matching", "max_independent_vertex_set",
           "min_spanning_tree", "random_spanning_tree", "dominator_tree",
           "topological_sort", "transitive_closure", "tsp_tour",
           "sequential_vertex_coloring", "label_components",
           "label_largest_component", "label_biconnected_components",
           "label_out_component", "vertex_percolation", "edge_percolation",
           "kcore_decomposition", "shortest_distance", "shortest_path",
           "all_shortest_paths", "all_predecessors", "all_paths",
           "all_circuits", "pseudo_diameter", "is_bipartite", "is_DAG",
           "is_planar", "make_maximal_planar", "similarity", "vertex_similarity",
           "edge_reciprocity"]

[docs]def similarity(g1, g2, eweight1=None, eweight2=None, label1=None, label2=None, norm=True, distance=False): r"""Return the adjacency similarity between the two graphs. Parameters ---------- g1 : :class:`~graph_tool.Graph` First graph to be compared. g2 : :class:`~graph_tool.Graph` Second graph to be compared. eweight1 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Edge weights for the first graph to be used in comparison. eweight2 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Edge weights for the second graph to be used in comparison. label1 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex labels for the first graph to be used in comparison. If not supplied, the vertex indexes are used. label2 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex labels for the second graph to be used in comparison. If not supplied, the vertex indexes are used. norm : bool (optional, default: ``True``) If ``True``, the returned value is normalized by the total number of edges. distance : bool (optional, default: ``False``) If ``True``, the complementary value is returned, i.e. the distance between the two graphs. Returns ------- similarity : float Adjacency similarity value. Notes ----- The adjacency similarity is the sum of equal entries in the adjacency matrix, given a vertex ordering determined by the vertex labels. In other words, it counts the number of edges which have the same source and target labels in both graphs. More specifically, it is defined as: .. math:: S(\boldsymbol A_1, \boldsymbol A_2) = E - d(\boldsymbol A_1, \boldsymbol A_2) where .. math:: d(\boldsymbol A_1, \boldsymbol A_2) = \sum_{i<j} |A_{ij}^{(1)} - A_{ij}^{(2)}| is the distance between graphs, and :math:`E=\sum_{i<j}|A_{ij}^{(1)}| + |A_{ij}^{(2)}|`. This definition holds for undirected graphs, otherwise the sums go over all directed pairs. If weights are provided, the weighted adjacency matrix is used. If ``norm == True`` the value returned is :math:`S(\boldsymbol A_1, \boldsymbol A_2) / E`. The algorithm runs with complexity :math:`O(E_1 + V_1 + E_2 + V_2)`. If enabled during compilation, and the vertex labels are integers, this algorithm runs in parallel. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: (3,3)) >>> u = g.copy() >>> gt.similarity(u, g) 1.0 >>> gt.random_rewire(u) 22 >>> gt.similarity(u, g) 0.04666666666666667 """ if label1 is None: label1 = g1.vertex_index if label2 is None: label2 = g2.vertex_index _check_prop_scalar(label1, name="label1") _check_prop_scalar(label2, name="label2") if label1.value_type() != label2.value_type(): try: label2 = label2.copy(label1.value_type()) except ValueError: label1 = label1.copy(label2.value_type()) if eweight1 is None and eweight1 is None: ew1 = ew2 = libcore.any() else: if eweight1 is None: eweight1 = g1.new_ep(eweight2.value_type(), 1) if eweight2 is None: eweight2 = g2.new_ep(eweight1.value_type(), 1) _check_prop_scalar(eweight1, name="eweight1") _check_prop_scalar(eweight2, name="eweight2") if eweight1.value_type() != eweight2.value_type(): try: eweight2 = eweight2.copy(eweight1.value_type()) except ValueError: eweight1 = eweight1.copy(eweight2.value_type()) ew1 = _prop("e", g1, eweight1) ew2 = _prop("e", g2, eweight2) if label1.is_writable() or label2.is_writable(): s = libgraph_tool_topology.\ similarity(g1._Graph__graph, g2._Graph__graph, ew1, ew2, _prop("v", g1, label1), _prop("v", g2, label2)) else: s = libgraph_tool_topology.\ similarity_fast(g1._Graph__graph, g2._Graph__graph, ew1, ew2, _prop("v", g1, label1), _prop("v", g2, label2)) if not g1.is_directed() or not g2.is_directed(): s /= 2 if eweight1 is None and eweight1 is None: E = g1.num_edges() + g2.num_edges() else: E = float(abs(eweight1.fa).sum() + abs(eweight2.fa).sum()) if not distance: s = E - s if norm: return s / E return s
[docs]@_limit_args({"sim_type": ["dice", "jaccard", "inv-log-weight"]}) def vertex_similarity(g, sim_type="jaccard", vertex_pairs=None, self_loops=True, sim_map=None): r"""Return the similarity between pairs of vertices. Parameters ---------- g : :class:`~graph_tool.Graph` The graph to be used. sim_type : ``str`` (optional, default: ``"jaccard"``) Type of similarity to use. This must be one of ``"dice"``, ``"jaccard"`` or ``"inv-log-weight"``. vertex_pairs : iterable of pairs of integers (optional, default: ``None``) Pairs of vertices to compute the similarity. If omitted, all pairs will be considered. self_loops : bool (optional, default: ``True``) If ``True``, vertices will be considered adjacent to themselves for the purpose of the similarity computation. sim_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) If provided, and ``vertex_pairs is None``, the vertex similarities will be stored in this vector-valued property. Otherwise, a new one will be created. Returns ------- similarities : :class:`numpy.ndarray` or :class:`~graph_tool.PropertyMap` If ``vertex_pairs`` was supplied, this will be a :class:`numpy.ndarray` with the corresponding similarities, otherwise it will be a vector-valued vertex :class:`~graph_tool.PropertyMap`, with the similarities to all other vertices. Notes ----- According to ``sim_type``, this function computes the following similarities: ``sim_type == "dice"`` The Sørensen–Dice similarity [sorensen-dice]_ is twice the number of common neighbors between two vertices divided by the sum of their degrees. ``sim_type == "jaccard"`` The Jaccard similarity [jaccard]_ is the number of common neighbors between two vertices divided by the size of the set of all neighbors to both vertices. ``sim_type == "inv-log-weight"`` The inverse log weighted similarity [adamic-friends-2003]_ is the sum of the weights of common neighbors between two vertices, where the weights are computed as :math:`1/\log(k)`, with :math:`k` being the degree of the vertex. For directed graphs, only out-neighbors are considered in the above algorthms (for "inv-log-weight", the in-degrees are used to compute the weights). To use the in-neighbors instead, a :class:`~graph_tool.GraphView` should be used to reverse the graph, e.g. ``vertex_similarity(GraphView(g, reversed=True))``. The algorithm runs with complexity :math:`O(\left<k\right>N^2)` if ``vertex_pairs is None``, otherwise with :math:`O(\left<k\right>P)` where :math:`P` is the length of ``vertex_pairs``. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testcode:: :hide: import matplotlib >>> g = gt.collection.data["polbooks"] >>> s = gt.vertex_similarity(g, "jaccard") >>> color = g.new_vp("double") >>> color.a = s[0].a >>> gt.graph_draw(g, pos=g.vp.pos, vertex_text=g.vertex_index, ... vertex_color=color, vertex_fill_color=color, ... vcmap=matplotlib.cm.inferno, ... output="polbooks-jaccard.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, pos=g.vp.pos, vertex_text=g.vertex_index, vertex_color=color, vertex_fill_color=color, vcmap=matplotlib.cm.inferno, output="polbooks-jaccard.png") .. figure:: polbooks-jaccard.* :align: center Jaccard similarities to vertex ``0`` in a political books network. References ---------- .. [sorensen-dice] https://en.wikipedia.org/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient .. [jaccard] https://en.wikipedia.org/wiki/Jaccard_index .. [adamic-friends-2003] Lada A. Adamic and Eytan Adar, "Friends and neighbors on the Web", Social Networks Volume 25, Issue 3, Pages 211–230 (2003) :doi:`10.1016/S0378-8733(03)00009-1` .. [liben-nowell-link-prediction-2007] David Liben-Nowell and Jon Kleinberg, "The link-prediction problem for social networks", Journal of the American Society for Information Science and Technology, Volume 58, Issue 7, pages 1019–1031 (2007), :doi:`10.1002/asi.20591` """ if vertex_pairs is None: if sim_map is None: s = g.new_vp("vector<double>") else: s = sim_map if sim_type == "dice": libgraph_tool_topology.dice_similarity(g._Graph__graph, _prop("v", g, s), self_loops) elif sim_type == "jaccard": libgraph_tool_topology.jaccard_similarity(g._Graph__graph, _prop("v", g, s), self_loops) elif sim_type == "inv-log-weight": libgraph_tool_topology.inv_log_weight_similarity(g._Graph__graph, _prop("v", g, s)) else: vertex_pairs = numpy.asarray(vertex_pairs, dtype="int64") s = numpy.zeros(vertex_pairs.shape[0], dtype="double") if sim_type == "dice": libgraph_tool_topology.dice_similarity_pairs(g._Graph__graph, vertex_pairs, s, self_loops) elif sim_type == "jaccard": libgraph_tool_topology.jaccard_similarity_pairs(g._Graph__graph, vertex_pairs, s, self_loops) elif sim_type == "inv-log-weight": libgraph_tool_topology.\ inv_log_weight_similarity_pairs(g._Graph__graph, vertex_pairs, s) return s
[docs]def isomorphism(g1, g2, vertex_inv1=None, vertex_inv2=None, isomap=False): r"""Check whether two graphs are isomorphic. Parameters ---------- g1 : :class:`~graph_tool.Graph` First graph. g2 : :class:`~graph_tool.Graph` Second graph. vertex_inv1 : :class:`~graph_tool.PropertyMap` (optional, default: `None`) Vertex invariant of the first graph. Only vertices with with the same invariants are considered in the isomorphism. vertex_inv2 : :class:`~graph_tool.PropertyMap` (optional, default: `None`) Vertex invariant of the second graph. Only vertices with with the same invariants are considered in the isomorphism. isomap : ``bool`` (optional, default: ``False``) If ``True``, a vertex :class:`~graph_tool.PropertyMap` with the isomorphism mapping is returned as well. Returns ------- is_isomorphism : ``bool`` ``True`` if both graphs are isomorphic, otherwise ``False``. isomap : :class:`~graph_tool.PropertyMap` Isomorphism mapping corresponding to a property map belonging to the first graph which maps its vertices to their corresponding vertices of the second graph. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: (3,3)) >>> g2 = gt.Graph(g) >>> gt.isomorphism(g, g2) True >>> g.add_edge(g.vertex(0), g.vertex(1)) <...> >>> gt.isomorphism(g, g2) False """ imap = g1.new_vertex_property("int32_t") if vertex_inv1 is None: vertex_inv1 = g1.degree_property_map("total").copy("int64_t") else: vertex_inv1 = vertex_inv1.copy("int64_t") d = g1.degree_property_map("total") vertex_inv1.fa += (vertex_inv1.fa.max() + 1) * d.a if vertex_inv2 is None: vertex_inv2 = g2.degree_property_map("total").copy("int64_t") else: vertex_inv2 = vertex_inv2.copy("int64_t") d = g2.degree_property_map("total") vertex_inv2.fa += (vertex_inv2.fa.max() + 1) * d.a inv_max = max(vertex_inv1.fa.max(),vertex_inv2.fa.max()) + 1 l1 = label_self_loops(g1, mark_only=True) if l1.fa.max() > 0: g1 = GraphView(g1, efilt=1 - l1.fa) l2 = label_self_loops(g2, mark_only=True) if l2.fa.max() > 0: g2 = GraphView(g2, efilt=1 - l2.fa) iso = libgraph_tool_topology.\ check_isomorphism(g1._Graph__graph, g2._Graph__graph, _prop("v", g1, vertex_inv1), _prop("v", g2, vertex_inv2), inv_max, _prop("v", g1, imap)) if isomap: return iso, imap else: return iso
[docs]def subgraph_isomorphism(sub, g, max_n=0, vertex_label=None, edge_label=None, induced=False, subgraph=True, generator=False): r"""Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n` subgraphs, if `max_n > 0`). Parameters ---------- sub : :class:`~graph_tool.Graph` Subgraph for which to be searched. g : :class:`~graph_tool.Graph` Graph in which the search is performed. max_n : int (optional, default: ``0``) Maximum number of matches to find. If `max_n == 0`, all matches are found. vertex_label : pair of :class:`~graph_tool.PropertyMap` (optional, default: ``None``) If provided, this should be a pair of :class:`~graph_tool.PropertyMap` objects, belonging to ``sub`` and ``g`` (in this order), which specify vertex labels which should match, in addition to the topological isomorphism. edge_label : pair of :class:`~graph_tool.PropertyMap` (optional, default: ``None``) If provided, this should be a pair of :class:`~graph_tool.PropertyMap` objects, belonging to ``sub`` and ``g`` (in this order), which specify edge labels which should match, in addition to the topological isomorphism. induced : bool (optional, default: ``False``) If ``True``, only node-induced subgraphs are found. subgraph : bool (optional, default: ``True``) If ``False``, all non-subgraph isomorphisms between `sub` and `g` are found. generator : bool (optional, default: ``False``) If ``True``, a generator will be returned, instead of a list. This is useful if the number of isomorphisms is too large to store in memory. If ``generator == True``, the option ``max_n`` is ignored. Returns ------- vertex_maps : list (or generator) of :class:`~graph_tool.PropertyMap` objects List (or generator) containing vertex property map objects which indicate different isomorphism mappings. The property maps vertices in `sub` to the corresponding vertex index in `g`. Notes ----- The implementation is based on the VF2 algorithm, introduced by Cordella et al. [cordella-improved-2001]_ [cordella-subgraph-2004]_. The spatial complexity is of order :math:`O(V)`, where :math:`V` is the (maximum) number of vertices of the two graphs. Time complexity is :math:`O(V^2)` in the best case and :math:`O(V!\times V)` in the worst case. Examples -------- >>> from numpy.random import poisson >>> g = gt.complete_graph(30) >>> sub = gt.complete_graph(10) >>> vm = gt.subgraph_isomorphism(sub, g, max_n=100) >>> print(len(vm)) 100 >>> for i in range(len(vm)): ... g.set_vertex_filter(None) ... g.set_edge_filter(None) ... vmask, emask = gt.mark_subgraph(g, sub, vm[i]) ... g.set_vertex_filter(vmask) ... g.set_edge_filter(emask) ... assert gt.isomorphism(g, sub) >>> g.set_vertex_filter(None) >>> g.set_edge_filter(None) >>> ewidth = g.copy_property(emask, value_type="double") >>> ewidth.a += 0.5 >>> ewidth.a *= 2 >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask, ... edge_pen_width=ewidth, output_size=(200, 200), ... output="subgraph-iso-embed.pdf") <...> >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask, edge_pen_width=ewidth, output_size=(200, 200), output="subgraph-iso-embed.png") gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.png") .. image:: subgraph-iso.* .. image:: subgraph-iso-embed.* **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph. References ---------- .. [cordella-improved-2001] L. P. Cordella, P. Foggia, C. Sansone, and M. Vento, "An improved algorithm for matching large graphs.", 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, pp. 149-159, Cuen, 2001. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.101.5342 .. [cordella-subgraph-2004] L. P. Cordella, P. Foggia, C. Sansone, and M. Vento, "A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs.", IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 10, pp. 1367-1372, 2004. :doi:`10.1109/TPAMI.2004.75` .. [boost-subgraph-iso] http://www.boost.org/libs/graph/doc/vf2_sub_graph_iso.html .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem """ if sub.num_vertices() == 0: raise ValueError("Cannot search for an empty subgraph.") if vertex_label is None: vertex_label = (None, None) elif vertex_label[0].value_type() != vertex_label[1].value_type(): raise ValueError("Both vertex label property maps must be of the same type!") elif vertex_label[0].value_type() != "int64_t": vertex_label = perfect_prop_hash(vertex_label, htype="int64_t") if edge_label is None: edge_label = (None, None) elif edge_label[0].value_type() != edge_label[1].value_type(): raise ValueError("Both edge label property maps must be of the same type!") elif edge_label[0].value_type() != "int64_t": edge_label = perfect_prop_hash(edge_label, htype="int64_t") vmaps = libgraph_tool_topology.\ subgraph_isomorphism(sub._Graph__graph, g._Graph__graph, _prop("v", sub, vertex_label[0]), _prop("v", g, vertex_label[1]), _prop("e", sub, edge_label[0]), _prop("e", g, edge_label[1]), max_n, induced, not subgraph, generator) if generator: return (PropertyMap(vmap, sub, "v") for vmap in vmaps) else: return [PropertyMap(vmap, sub, "v") for vmap in vmaps]
[docs]def mark_subgraph(g, sub, vmap, vmask=None, emask=None): r""" Mark a given subgraph `sub` on the graph `g`. The mapping must be provided by the `vmap` and `emap` parameters, which map vertices/edges of `sub` to indexes of the corresponding vertices/edges in `g`. This returns a vertex and an edge property map, with value type 'bool', indicating whether or not a vertex/edge in `g` corresponds to the subgraph `sub`. """ if vmask is None: vmask = g.new_vertex_property("bool") if emask is None: emask = g.new_edge_property("bool") vmask.a = False emask.a = False for v in sub.vertices(): w = g.vertex(vmap[v]) vmask[w] = True us = set([g.vertex(vmap[x]) for x in v.out_neighbors()]) for ew in w.out_edges(): if ew.target() in us: emask[ew] = True return vmask, emask
[docs]def min_spanning_tree(g, weights=None, root=None, tree_map=None): """ Return the minimum spanning tree of a given graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`) The edge weights. If provided, the minimum spanning tree will minimize the edge weights. root : :class:`~graph_tool.Vertex` (optional, default: `None`) Root of the minimum spanning tree. If this is provided, Prim's algorithm is used. Otherwise, Kruskal's algorithm is used. tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`) If provided, the edge tree map will be written in this property map. Returns ------- tree_map : :class:`~graph_tool.PropertyMap` Edge property map with mark the tree edges: 1 for tree edge, 0 otherwise. Notes ----- The algorithm runs with :math:`O(E\log E)` complexity, or :math:`O(E\log V)` if `root` is specified. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> from numpy.random import random >>> g, pos = gt.triangulation(random((400, 2)) * 10, type="delaunay") >>> weight = g.new_edge_property("double") >>> for e in g.edges(): ... weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a) >>> tree = gt.min_spanning_tree(g, weights=weight) >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf") <...> >>> u = gt.GraphView(g, efilt=tree) >>> gt.graph_draw(u, pos=pos, output="triang_min_span_tree.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, pos=pos, output="triang_orig.png") gt.graph_draw(u, pos=pos, output="triang_min_span_tree.png") .. image:: triang_orig.* :width: 400px .. image:: triang_min_span_tree.* :width: 400px *Left:* Original graph, *Right:* The minimum spanning tree. References ---------- .. [kruskal-shortest-1956] J. B. Kruskal. "On the shortest spanning subtree of a graph and the traveling salesman problem", In Proceedings of the American Mathematical Society, volume 7, pages 48-50, 1956. :doi:`10.1090/S0002-9939-1956-0078686-7` .. [prim-shortest-1957] R. Prim. "Shortest connection networks and some generalizations", Bell System Technical Journal, 36:1389-1401, 1957. .. [boost-mst] http://www.boost.org/libs/graph/doc/graph_theory_review.html#sec:minimum-spanning-tree .. [mst-wiki] http://en.wikipedia.org/wiki/Minimum_spanning_tree """ if tree_map is None: tree_map = g.new_edge_property("bool") if tree_map.value_type() != "bool": raise ValueError("edge property 'tree_map' must be of value type bool.") u = GraphView(g, directed=False) if root is None: libgraph_tool_topology.\ get_kruskal_spanning_tree(u._Graph__graph, _prop("e", g, weights), _prop("e", g, tree_map)) else: libgraph_tool_topology.\ get_prim_spanning_tree(u._Graph__graph, int(root), _prop("e", g, weights), _prop("e", g, tree_map)) return tree_map
[docs]def random_spanning_tree(g, weights=None, root=None, tree_map=None): r"""Return a random spanning tree of a given graph, which can be directed or undirected. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`) The edge weights. If provided, the probability of a particular spanning tree being selected is the product of its edge weights. root : :class:`~graph_tool.Vertex` (optional, default: `None`) Root of the spanning tree. If not provided, it will be selected randomly. tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`) If provided, the edge tree map will be written in this property map. Returns ------- tree_map : :class:`~graph_tool.PropertyMap` Edge property map with mark the tree edges: 1 for tree edge, 0 otherwise. Notes ----- The running time for this algorithm is :math:`O(\tau)`, with :math:`\tau` being the mean hitting time of a random walk on the graph. In the worse case, we have :math:`\tau \sim O(V^3)`, with :math:`V` being the number of vertices in the graph. However, in much more typical cases (e.g. sparse random graphs) the running time is simply :math:`O(V)`. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> from numpy.random import random >>> g, pos = gt.triangulation(random((400, 2)), type="delaunay") >>> weight = g.new_edge_property("double") >>> for e in g.edges(): ... weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a) >>> tree = gt.random_spanning_tree(g, weights=weight) >>> tree2 = gt.random_spanning_tree(g, weights=weight) >>> gt.graph_draw(g, pos=pos, output="rtriang_orig.pdf") <...> >>> u = gt.GraphView(g, efilt=tree) >>> gt.graph_draw(u, pos=pos, output="triang_random_span_tree.pdf") <...> >>> u2 = gt.GraphView(g, efilt=tree2) >>> gt.graph_draw(u2, pos=pos, output="triang_random_span_tree2.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, pos=pos, output="rtriang_orig.png") gt.graph_draw(u, pos=pos, output="triang_random_span_tree.png") gt.graph_draw(u2, pos=pos, output="triang_random_span_tree2.png") .. image:: rtriang_orig.* :width: 300px .. image:: triang_random_span_tree.* :width: 300px .. image:: triang_random_span_tree2.* :width: 300px *Left:* Original graph, *Middle:* A random spanning tree, *Right:* Another random spanning tree References ---------- .. [wilson-generating-1996] David Bruce Wilson, "Generating random spanning trees more quickly than the cover time", Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, Pages 296-303, ACM New York, 1996, :doi:`10.1145/237814.237880` .. [boost-rst] http://www.boost.org/libs/graph/doc/random_spanning_tree.html """ if tree_map is None: tree_map = g.new_edge_property("bool") if tree_map.value_type() != "bool": raise ValueError("edge property 'tree_map' must be of value type bool.") if root is None: root = g.vertex(numpy.random.randint(0, g.num_vertices()), use_index=False) # we need to restrict ourselves to the in-component of root l = label_out_component(GraphView(g, reversed=True), root) u = GraphView(g, vfilt=l) if u.num_vertices() != g.num_vertices(): raise ValueError("There must be a path from all vertices to the root vertex: %d" % int(root) ) libgraph_tool_topology.\ random_spanning_tree(g._Graph__graph, int(root), _prop("e", g, weights), _prop("e", g, tree_map), _get_rng()) return tree_map
[docs]def dominator_tree(g, root, dom_map=None): """Return a vertex property map the dominator vertices for each vertex. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. root : :class:`~graph_tool.Vertex` The root vertex. dom_map : :class:`~graph_tool.PropertyMap` (optional, default: None) If provided, the dominator map will be written in this property map. Returns ------- dom_map : :class:`~graph_tool.PropertyMap` The dominator map. It contains for each vertex, the index of its dominator vertex. Notes ----- A vertex u dominates a vertex v, if every path of directed graph from the entry to v must go through u. The algorithm runs with :math:`O((V+E)\log (V+E))` complexity. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: (2, 2)) >>> tree = gt.min_spanning_tree(g) >>> g.set_edge_filter(tree) >>> root = [v for v in g.vertices() if v.in_degree() == 0] >>> dom = gt.dominator_tree(g, root[0]) >>> print(dom.a) [ 0 0 0 0 0 0 0 74 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 64 67 0 0 67 0 0 74 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0] References ---------- .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm """ if dom_map is None: dom_map = g.new_vertex_property("int32_t") if dom_map.value_type() != "int32_t": raise ValueError("vertex property 'dom_map' must be of value type" + " int32_t.") if not g.is_directed(): raise ValueError("dominator tree requires a directed graph.") libgraph_tool_topology.\ dominator_tree(g._Graph__graph, int(root), _prop("v", g, dom_map)) return dom_map
[docs]def topological_sort(g): """ Return the topological sort of the given graph. It is returned as an array of vertex indexes, in the sort order. Notes ----- The topological sort algorithm creates a linear ordering of the vertices such that if edge (u,v) appears in the graph, then u comes before v in the ordering. The graph must be a directed acyclic graph (DAG). The time complexity is :math:`O(V + E)`. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(30, lambda: (3, 3)) >>> tree = gt.min_spanning_tree(g) >>> g.set_edge_filter(tree) >>> sort = gt.topological_sort(g) >>> print(sort) [28 26 29 27 23 22 18 17 16 20 21 15 12 11 10 25 14 9 8 7 5 3 2 24 4 6 1 0 19 13] References ---------- .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting """ topological_order = Vector_int32_t() is_DAG = libgraph_tool_topology.\ topological_sort(g._Graph__graph, topological_order) if not is_DAG: raise ValueError("Graph is not a directed acylic graph (DAG)."); return topological_order.a[::-1].copy()
[docs]def transitive_closure(g): """Return the transitive closure graph of g. Notes ----- The transitive closure of a graph G = (V,E) is a graph G* = (V,E*) such that E* contains an edge (u,v) if and only if G contains a path (of at least one edge) from u to v. The transitive_closure() function transforms the input graph g into the transitive closure graph tc. The time complexity (worst-case) is :math:`O(VE)`. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(30, lambda: (3, 3)) >>> tc = gt.transitive_closure(g) References ---------- .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure """ if not g.is_directed(): raise ValueError("graph must be directed for transitive closure.") tg = Graph() libgraph_tool_topology.transitive_closure(g._Graph__graph, tg._Graph__graph) return tg
[docs]def label_components(g, vprop=None, directed=None, attractors=False): """ Label the components to which each vertex in the graph belongs. If the graph is directed, it finds the strongly connected components. A property map with the component labels is returned, together with an histogram of component labels. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex property to store the component labels. If none is supplied, one is created. directed : bool (optional, default: ``None``) Treat graph as directed or not, independently of its actual directionality. attractors : bool (optional, default: ``False``) If ``True``, and the graph is directed, an additional array with Boolean values is returned, specifying if the strongly connected components are attractors or not. Returns ------- comp : :class:`~graph_tool.PropertyMap` Vertex property map with component labels. hist : :class:`~numpy.ndarray` Histogram of component labels. is_attractor : :class:`~numpy.ndarray` A Boolean array specifying if the strongly connected components are attractors or not. This returned only if ``attractors == True``, and the graph is directed. Notes ----- The components are arbitrarily labeled from 0 to N-1, where N is the total number of components. The algorithm runs in :math:`O(V + E)` time. Examples -------- .. testcode:: :hide: numpy.random.seed(43) gt.seed_rng(43) >>> g = gt.random_graph(100, lambda: (poisson(2), poisson(2))) >>> comp, hist, is_attractor = gt.label_components(g, attractors=True) >>> print(comp.a) [ 9 9 9 9 10 1 9 11 12 9 9 9 9 9 9 13 9 9 9 0 9 9 16 9 9 3 9 9 4 17 9 9 18 9 9 19 20 9 9 9 14 5 9 9 6 9 9 9 21 9 9 9 9 9 9 9 9 9 9 9 9 9 9 2 9 8 9 22 15 9 9 9 9 9 23 25 9 9 26 27 28 29 30 9 9 9 9 9 9 31 9 9 9 9 9 32 9 9 7 24] >>> print(hist) [ 1 1 1 1 1 1 1 1 1 68 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] >>> print(is_attractor) [ True True True True True True True True True False True False False False False False False False False False False False False False False False False False True False True False False] """ if vprop is None: vprop = g.new_vertex_property("int32_t") _check_prop_writable(vprop, name="vprop") _check_prop_scalar(vprop, name="vprop") if directed is not None: g = GraphView(g, directed=directed) hist = libgraph_tool_topology.\ label_components(g._Graph__graph, _prop("v", g, vprop)) if attractors and g.is_directed() and directed != False: is_attractor = numpy.ones(len(hist), dtype="bool") libgraph_tool_topology.\ label_attractors(g._Graph__graph, _prop("v", g, vprop), is_attractor) return vprop, hist, is_attractor else: return vprop, hist
[docs]def label_largest_component(g, directed=None): """ Label the largest component in the graph. If the graph is directed, then the largest strongly connected component is labelled. A property map with a boolean label is returned. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. directed : bool (optional, default:None) Treat graph as directed or not, independently of its actual directionality. Returns ------- comp : :class:`~graph_tool.PropertyMap` Boolean vertex property map which labels the largest component. Notes ----- The algorithm runs in :math:`O(V + E)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: poisson(1), directed=False) >>> l = gt.label_largest_component(g) >>> print(l.a) [0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0] >>> u = gt.GraphView(g, vfilt=l) # extract the largest component as a graph >>> print(u.num_vertices()) 18 """ label = g.new_vertex_property("bool") c, h = label_components(g, directed=directed) vfilt, inv = g.get_vertex_filter() label.fa = c.fa == h.argmax() return label
[docs]def label_out_component(g, root, label=None): """ Label the out-component (or simply the component for undirected graphs) of a root vertex. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. root : :class:`~graph_tool.Vertex` The root vertex. label : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) If provided, this must be an initialized Boolean vertex property map where the out-component will be labeled. Returns ------- label : :class:`~graph_tool.PropertyMap` Boolean vertex property map which labels the out-component. Notes ----- The algorithm runs in :math:`O(V + E)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: poisson(2.2), directed=False) >>> l = gt.label_out_component(g, g.vertex(2)) >>> print(l.a) [1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0] The in-component can be obtained by reversing the graph. >>> l = gt.label_out_component(gt.GraphView(g, reversed=True, directed=True), ... g.vertex(1)) >>> print(l.a) [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] """ if label is None: label = g.new_vertex_property("bool") elif label.value_type() != "bool": raise ValueError("value type of `label` must be `bool`, not %s" % label.value_type()) libgraph_tool_topology.\ label_out_component(g._Graph__graph, int(root), _prop("v", g, label)) return label
[docs]def label_biconnected_components(g, eprop=None, vprop=None): """ Label the edges of biconnected components, and the vertices which are articulation points. An edge property map with the component labels is returned, together a boolean vertex map marking the articulation points, and an histogram of component labels. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. eprop : :class:`~graph_tool.PropertyMap` (optional, default: None) Edge property to label the biconnected components. vprop : :class:`~graph_tool.PropertyMap` (optional, default: None) Vertex property to mark the articulation points. If none is supplied, one is created. Returns ------- bicomp : :class:`~graph_tool.PropertyMap` Edge property map with the biconnected component labels. articulation : :class:`~graph_tool.PropertyMap` Boolean vertex property map which has value 1 for each vertex which is an articulation point, and zero otherwise. nc : int Number of biconnected components. Notes ----- A connected graph is biconnected if the removal of any single vertex (and all edges incident on that vertex) can not disconnect the graph. More generally, the biconnected components of a graph are the maximal subsets of vertices such that the removal of a vertex from a particular component will not disconnect the component. Unlike connected components, vertices may belong to multiple biconnected components: those vertices that belong to more than one biconnected component are called "articulation points" or, equivalently, "cut vertices". Articulation points are vertices whose removal would increase the number of connected components in the graph. Thus, a graph without articulation points is biconnected. Vertices can be present in multiple biconnected components, but each edge can only be contained in a single biconnected component. The algorithm runs in :math:`O(V + E)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(100, lambda: poisson(2), directed=False) >>> comp, art, hist = gt.label_biconnected_components(g) >>> print(comp.a) [26 26 26 26 26 26 26 26 19 25 26 26 23 26 26 26 26 6 26 24 18 26 26 13 26 26 26 26 26 26 26 26 26 26 26 16 29 26 26 26 26 26 26 15 26 26 26 26 26 0 26 26 12 2 26 26 26 26 26 26 26 26 9 3 26 28 26 26 8 26 4 26 26 26 14 26 26 26 26 30 11 26 26 26 20 26 26 27 26 33 26 22 17 7 5 32 21 26 1 10 31] >>> print(art.a) [1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0] >>> print(hist) [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 68 1 1 1 1 1 1 1] """ if vprop is None: vprop = g.new_vertex_property("bool") if eprop is None: eprop = g.new_edge_property("int32_t") _check_prop_writable(vprop, name="vprop") _check_prop_scalar(vprop, name="vprop") _check_prop_writable(eprop, name="eprop") _check_prop_scalar(eprop, name="eprop") g = GraphView(g, directed=False) hist = libgraph_tool_topology.\ label_biconnected_components(g._Graph__graph, _prop("e", g, eprop), _prop("v", g, vprop)) return eprop, vprop, hist
[docs]def vertex_percolation(g, vertices): """Compute the size of the largest component as vertices are (virtually) removed from the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. vertices : :class:`numpy.ndarray` or iterable of ints List of vertices in reversed order of removal. Returns ------- size : :class:`numpy.ndarray` Size of the largest component prior to removal of each vertex. comp : :class:`~graph_tool.PropertyMap` Vertex property map with component labels. Notes ----- The algorithm runs in :math:`O(V + E)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(10000, lambda: geometric(1./4) + 1, directed=False) >>> vertices = sorted([v for v in g.vertices()], key=lambda v: v.out_degree()) >>> sizes, comp = gt.vertex_percolation(g, vertices) >>> numpy.random.shuffle(vertices) >>> sizes2, comp = gt.vertex_percolation(g, vertices) >>> figure() <...> >>> plot(sizes, label="Targeted") [...] >>> plot(sizes2, label="Random") [...] >>> xlabel("Vertices remaining") Text(...) >>> ylabel("Size of largest component") Text(...) >>> legend(loc="lower right") <...> >>> savefig("vertex-percolation.svg") .. figure:: vertex-percolation.* :align: center Targeted and random vertex percolation of a random graph with an exponential degree distribution. References ---------- .. [newman-ziff] M. E. J. Newman, R. M. Ziff, "A fast Monte Carlo algorithm for site or bond percolation", Phys. Rev. E 64, 016706 (2001) :doi:`10.1103/PhysRevE.64.016706`, :arxiv:`cond-mat/0101295` """ vertices = numpy.asarray(vertices, dtype="uint64") tree = g.vertex_index.copy("int64_t") size = g.new_vertex_property("int64_t", 1) visited = g.new_vertex_property("bool", False) max_size = numpy.zeros(len(vertices), dtype="uint64") u = GraphView(g, directed=False) libgraph_tool_topology.\ percolate_vertex(u._Graph__graph, _prop("v", u, tree), _prop("v", u, size), _prop("v", u, visited), vertices, max_size) return max_size, tree
[docs]def edge_percolation(g, edges): """Compute the size of the largest component as edges are (virtually) removed from the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. edges : :class:`numpy.ndarray` or iterable of pairs of ints List of edges in reversed order of removal. If the type is :class:`numpy.ndarray`, it should have a shape ``(E, 2)``, where ``E`` is the number of edges, such that ``edges[i,0]`` and ``edges[i,1]`` are the both endpoints of edge ``i``. Returns ------- size : :class:`numpy.ndarray` Size of the largest component prior to removal of each edge. comp : :class:`~graph_tool.PropertyMap` Vertex property map with component labels. Notes ----- The algorithm runs in :math:`O(E)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(10000, lambda: geometric(1./4) + 1, directed=False) >>> edges = sorted([(e.source(), e.target()) for e in g.edges()], ... key=lambda e: e[0].out_degree() * e[1].out_degree()) >>> sizes, comp = gt.edge_percolation(g, edges) >>> numpy.random.shuffle(edges) >>> sizes2, comp = gt.edge_percolation(g, edges) >>> figure() <...> >>> plot(sizes, label="Targeted") [...] >>> plot(sizes2, label="Random") [...] >>> xlabel("Edges remaining") Text(...) >>> ylabel("Size of largest component") Text(...) >>> legend(loc="lower right") <...> >>> savefig("edge-percolation.svg") .. figure:: edge-percolation.* :align: center Targeted and random edge percolation of a random graph with an exponential degree distribution. References ---------- .. [newman-ziff] M. E. J. Newman, R. M. Ziff, "A fast Monte Carlo algorithm for site or bond percolation", Phys. Rev. E 64, 016706 (2001) :doi:`10.1103/PhysRevE.64.016706`, :arxiv:`cond-mat/0101295` """ edges = numpy.asarray(edges, dtype="uint64") tree = g.vertex_index.copy("int64_t") size = g.new_vertex_property("int64_t", 1) max_size = numpy.zeros(len(edges), dtype="uint64") u = GraphView(g, directed=False) libgraph_tool_topology.\ percolate_edge(u._Graph__graph, _prop("v", u, tree), _prop("v", u, size), edges, max_size) return max_size, tree
[docs]def kcore_decomposition(g, vprop=None): """Perform a k-core decomposition of the given graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex property to store the decomposition. If ``None`` is supplied, one is created. Returns ------- kval : :class:`~graph_tool.PropertyMap` Vertex property map with the k-core decomposition, i.e. a given vertex v belongs to the ``kval[v]``-core. Notes ----- The k-core is a maximal set of vertices such that its induced subgraph only contains vertices with degree larger than or equal to k. For directed graphs, the degree is assumed to be the total (in + out) degree. The algorithm accepts graphs with parallel edges and self loops, in which case these edges contribute to the degree in the usual fashion. This algorithm is described in [batagelk-algorithm]_ and runs in :math:`O(V + E)` time. Examples -------- >>> g = gt.collection.data["netscience"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> kcore = gt.kcore_decomposition(g) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=kcore, vertex_text=kcore, output="netsci-kcore.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=kcore, vertex_text=kcore, output="netsci-kcore.png") .. figure:: netsci-kcore.* :align: center K-core decomposition of a network of network scientists. References ---------- .. [k-core] http://en.wikipedia.org/wiki/Degeneracy_%28graph_theory%29 .. [batagelk-algorithm] Vladimir Batagelj, Matjaž Zaveršnik, "Fast algorithms for determining (generalized) core groups in social networks", Advances in Data Analysis and Classification Volume 5, Issue 2, pp 129-145 (2011), :DOI:`10.1007/s11634-010-0079-y`, :arxiv:`cs/0310049` """ if vprop is None: vprop = g.new_vertex_property("int32_t") _check_prop_writable(vprop, name="vprop") _check_prop_scalar(vprop, name="vprop") libgraph_tool_topology.\ kcore_decomposition(g._Graph__graph, _prop("v", g, vprop)) return vprop
[docs]def shortest_distance(g, source=None, target=None, weights=None, negative_weights=False, max_dist=None, directed=None, dense=False, dist_map=None, pred_map=False, return_reached=False): """Calculate the distance from a source to a target vertex, or to of all vertices from a given source, or the all pairs shortest paths, if the source is not specified. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. source : :class:`~graph_tool.Vertex` (optional, default: ``None``) Source vertex of the search. If unspecified, the all pairs shortest distances are computed. target : :class:`~graph_tool.Vertex` or iterable of such objects (optional, default: ``None``) Target vertex (or vertices) of the search. If unspecified, the distance to all vertices from the source will be computed. weights : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) The edge weights. If provided, the shortest path will correspond to the minimal sum of weights. negative_weights : ``bool`` (optional, default: ``False``) If `True`, this will trigger the use of the Bellman-Ford algorithm. Ignored if ``source`` is ``None``. max_dist : scalar value (optional, default: ``None``) If specified, this limits the maximum distance of the vertices searched. This parameter has no effect if source is ``None``, or if `negative_weights=True`. directed : ``bool`` (optional, default:``None``) Treat graph as directed or not, independently of its actual directionality. dense : ``bool`` (optional, default: ``False``) If ``True``, and source is ``None``, the Floyd-Warshall algorithm is used, otherwise the Johnson algorithm is used. If source is not ``None``, this option has no effect. dist_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``) Vertex property to store the distances. If none is supplied, one is created. .. warning:: If this parameter is supplied, the user is responsible for initializing it to infinity. This can be done as: >>> dist_map = g.new_vp("double", numpy.inf) or >>> dist_map = g.new_vp("int32_t", numpy.iinfo(numpy.int32).max) depending on the distance type. pred_map : ``bool`` or :class:`~graph_tool.PropertyMap` (optional, default: ``False``) If ``True``, a vertex property map with the predecessors is returned. If a :class:`~graph_tool.PropertyMap` is passed, it must be of value ``int64_t`` and it will be used to store the predecessors. Ignored if ``source`` is ``None``. .. warning:: If a property map is supplied, the user is responsible for initializing to the identity map. This can be done as: >>> pred_map = g.vertex_index.copy() return_reached : ``bool`` (optional, default: ``False``) If ``True``, return an array of visited vertices. Returns ------- dist_map : :class:`~graph_tool.PropertyMap` or :class:`numpy.ndarray` Vertex property map with the distances from source. If ``source`` is ``None``, it will have a vector value type, with the distances to every vertex. If ``target`` is an iterable, instead of :class:`~graph_tool.PropertyMap`, this will be of type :class:`numpy.ndarray`, and contain only the distances to those specific targets. pred_map : :class:`~graph_tool.PropertyMap` (optional, if ``pred_map == True``) Vertex property map with the predecessors in the search tree. pred_map : :class:`numpy.ndarray` (optional, if ``return_reached == True``) Array containing vertices visited during the search. Notes ----- If a source is given, the distances are calculated with a breadth-first search (BFS) or Dijkstra's algorithm [dijkstra]_, if weights are given. If ``negative_weights == True``, the Bellman-Ford algorithm is used [bellman-ford]_, which accepts negative weights, as long as there are no negative loops. If source is not given, the distances are calculated with Johnson's algorithm [johnson-apsp]_. If dense=True, the Floyd-Warshall algorithm [floyd-warshall-apsp]_ is used instead. If there is no path between two vertices, the computed distance will correspond to the maximum value allowed by the value type of ``dist_map``, or ``inf`` in case of floating point types. If source is specified, the algorithm runs in :math:`O(V + E)` time, or :math:`O(V \log V)` if weights are given. If ``negative_weights == True``, the complexity is :math:`O(VE)`. If source is not specified, it runs in :math:`O(VE\log V)` time, or :math:`O(V^3)` if dense == True. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> from numpy.random import poisson >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> dist = gt.shortest_distance(g, source=g.vertex(0)) >>> print(dist.a) [ 0 4 5 6 2147483647 5 4 3 2147483647 4 4 6 6 5 5 4 4 2 5 1 2147483647 6 5 5 5 7 5 4 6 5 5 4 1 4 4 3 5 2147483647 5 1 2147483647 2 2 7 4 5 5 5 6 5 5 3 2 7 5 4 3 5 4 6 3 5 5 3 4 4 6 4 4 5 2147483647 2147483647 2 5 7 3 2147483647 2147483647 5 6 4 7 4 4 3 4 6 4 3 2147483647 5 6 3 4 6 5 5 3 4 5] >>> >>> dist = gt.shortest_distance(g) >>> print(dist[g.vertex(0)].a) [ 0 4 5 6 2147483647 5 4 3 2147483647 4 4 6 6 5 5 4 4 2 5 1 2147483647 6 5 5 5 7 5 4 6 5 5 4 1 4 4 3 5 2147483647 5 1 2147483647 2 2 7 4 5 5 5 6 5 5 3 2 7 5 4 3 5 4 6 3 5 5 3 4 4 6 4 4 5 2147483647 2147483647 2 5 7 3 2147483647 2147483647 5 6 4 7 4 4 3 4 6 4 3 2147483647 5 6 3 4 6 5 5 3 4 5] >>> dist = gt.shortest_distance(g, source=g.vertex(0), target=g.vertex(2)) >>> print(dist) 5 >>> dist = gt.shortest_distance(g, source=g.vertex(0), target=[g.vertex(2), g.vertex(6)]) >>> print(dist) [5 4] References ---------- .. [bfs] Edward Moore, "The shortest path through a maze", International Symposium on the Theory of Switching (1959), Harvard University Press. .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with graphs." Numerische Mathematik, 1:269-271, 1959. .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html .. [johnson-apsp] http://www.boost.org/libs/graph/doc/johnson_all_pairs_shortest.html .. [floyd-warshall-apsp] http://www.boost.org/libs/graph/doc/floyd_warshall_shortest.html .. [bellman-ford] http://www.boost.org/libs/graph/doc/bellman_ford_shortest.html """ tgtlist = False if isinstance(target, collections.Iterable): tgtlist = True target = numpy.asarray(target, dtype="int64") elif target is None: target = numpy.array([], dtype="int64") else: target = numpy.asarray([int(target)], dtype="int64") if weights is None: dist_type = 'int32_t' else: dist_type = weights.value_type() if dist_map is None: if source is not None: dist_map = g.new_vertex_property(dist_type) else: dist_map = g.new_vertex_property("vector<%s>" % dist_type) _check_prop_writable(dist_map, name="dist_map") if source is not None: _check_prop_scalar(dist_map, name="dist_map") else: _check_prop_vector(dist_map, name="dist_map") if max_dist is None: max_dist = 0 if directed is not None: u = GraphView(g, directed=directed) else: u = g if source is not None: if numpy.issubdtype(dist_map.a.dtype, numpy.integer): dist_map.fa = numpy.iinfo(dist_map.a.dtype).max else: dist_map.fa = numpy.inf if isinstance(pred_map, PropertyMap): pmap = pred_map if pmap.value_type() != "int64_t": raise ValueError("supplied pred_map must be of value type 'int64_t'") else: pmap = u.copy_property(u.vertex_index, value_type="int64_t") reached = libcore.Vector_size_t() libgraph_tool_topology.get_dists(u._Graph__graph, int(source), target, _prop("v", u, dist_map), _prop("e", u, weights), _prop("v", u, pmap), float(max_dist), negative_weights, reached) else: libgraph_tool_topology.get_all_dists(u._Graph__graph, _prop("v", u, dist_map), _prop("e", u, weights), dense) if source is not None and len(target) > 0: if len(target) == 1 and not tgtlist: dist_map = dist_map.a[target[0]] else: dist_map = numpy.array(dist_map.a[target]) if source is not None: if pred_map: ret = (dist_map, pmap) else: ret = (dist_map,) if return_reached: return ret + (numpy.asarray(reached.a.copy()),) else: if len(ret) == 1: return ret[0] return ret else: return dist_map
[docs]def shortest_path(g, source, target, weights=None, negative_weights=False, pred_map=None): """Return the shortest path from ``source`` to ``target``. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. source : :class:`~graph_tool.Vertex` Source vertex of the search. target : :class:`~graph_tool.Vertex` Target vertex of the search. weights : :class:`~graph_tool.PropertyMap` (optional, default: None) The edge weights. negative_weights : ``bool`` (optional, default: ``False``) If ``True``, this will trigger the use of the Bellman-Ford algorithm. pred_map : :class:`~graph_tool.PropertyMap` (optional, default: None) Vertex property map with the predecessors in the search tree. If this is provided, the shortest paths are not computed, and are obtained directly from this map. Returns ------- vertex_list : list of :class:`~graph_tool.Vertex` List of vertices from `source` to `target` in the shortest path. edge_list : list of :class:`~graph_tool.Edge` List of edges from `source` to `target` in the shortest path. Notes ----- The paths are computed with a breadth-first search (BFS) or Dijkstra's algorithm [dijkstra]_, if weights are given. If ``negative_weights == True``, the Bellman-Ford algorithm is used [bellman-ford]_, which accepts negative weights, as long as there are no negative loops. The algorithm runs in :math:`O(V + E)` time, or :math:`O(V \log V)` if weights are given. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(43) gt.seed_rng(43) >>> from numpy.random import poisson >>> g = gt.random_graph(300, lambda: (poisson(4), poisson(4))) >>> vlist, elist = gt.shortest_path(g, g.vertex(10), g.vertex(11)) >>> print([str(v) for v in vlist]) ['10', '131', '118', '207', '195', '11'] >>> print([str(e) for e in elist]) ['(10, 131)', '(131, 118)', '(118, 207)', '(207, 195)', '(195, 11)'] References ---------- .. [bfs] Edward Moore, "The shortest path through a maze", International Symposium on the Theory of Switching (1959), Harvard University Press .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with graphs." Numerische Mathematik, 1:269-271, 1959. .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html .. [bellman-ford] http://www.boost.org/libs/graph/doc/bellman_ford_shortest.html """ if pred_map is None: pred_map = shortest_distance(g, source, target, weights=weights, negative_weights=negative_weights, pred_map=True)[1] if pred_map[target] == int(target): # no path to target return [], [] vlist = [target] elist = [] if weights is not None: max_w = weights.a.max() + 1 else: max_w = None v = target while v != source: p = g.vertex(pred_map[v]) min_w = max_w pe = None s = None for e in v.in_edges() if g.is_directed() else v.out_edges(): s = e.source() if g.is_directed() else e.target() if s == p: if weights is not None: if weights[e] < min_w: min_w = weights[e] pe = e else: pe = e break elist.insert(0, pe) vlist.insert(0, p) v = p return vlist, elist
[docs]def all_predecessors(g, dist_map, pred_map, weights=None, epsilon=1e-8): """Return a property map with all possible predecessors in the search tree determined by ``dist_map`` and ``pred_map``. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. dist_map : :class:`~graph_tool.PropertyMap` Vertex property map with the distances from ``source`` to all other vertices. pred_map : :class:`~graph_tool.PropertyMap` (optional, default: None) Vertex property map with the predecessors in the search tree. weights : :class:`~graph_tool.PropertyMap` (optional, default: None) The edge weights. epsilon : `float` (optional, default: `1e-8`) Maximum relative difference between distances to be considered "equal", in case floating-point weights are used. Returns ------- all_preds_map : :class:`~graph_tool.PropertyMap` Vector-valued vertex property map with all possible predecessors in the search tree. """ preds = g.new_vertex_property("vector<int64_t>") libgraph_tool_topology.get_all_preds(g._Graph__graph, _prop("v", g, dist_map), _prop("v", g, pred_map), _prop("e", g, weights), _prop("v", g, preds), epsilon) return preds
[docs]def all_shortest_paths(g, source, target, weights=None, negative_weights=False, dist_map=None, pred_map=None, all_preds_map=None, epsilon=1e-8): """Return an iterator over all shortest paths from `source` to `target`. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. source : :class:`~graph_tool.Vertex` Source vertex of the search. target : :class:`~graph_tool.Vertex` Target vertex of the search. weights : :class:`~graph_tool.PropertyMap` (optional, default: None) The edge weights. negative_weights : ``bool`` (optional, default: ``False``) If ``True``, this will trigger the use of the Bellman-Ford algorithm. dist_map : :class:`~graph_tool.PropertyMap` (optional, default: None) Vertex property map with the distances from ``source`` to all other vertices. pred_map : :class:`~graph_tool.PropertyMap` (optional, default: None) Vertex property map with the predecessors in the search tree. If this is provided, the shortest paths are not computed, and are obtained directly from this map. all_preds_map : :class:`~graph_tool.PropertyMap` (optional, default: None) Vector-valued vertex property map with all possible predecessors in the search tree. If this is provided, the shortest paths are obtained directly from this map. epsilon : `float` (optional, default: `1e-8`) Maximum relative difference between distances to be considered "equal", in case floating-point weights are used. Returns ------- path_iterator : iterator over a sequence of integers Iterator over sequences of vertices from `source` to `target` in the shortest path. Notes ----- The paths are computed with a breadth-first search (BFS) or Dijkstra's algorithm [dijkstra]_, if weights are given. If ``negative_weights == True``, the Bellman-Ford algorithm is used [bellman-ford]_, which accepts negative weights, as long as there are no negative loops. If both ``dist_map`` and ``pred_map` are provided, the search is not actually performed. Examples -------- >>> g = gt.collection.data["pgp-strong-2009"] >>> for path in gt.all_shortest_paths(g, 92, 45): ... print(path) [ 92 107 2176 7027 26 21 45] [ 92 107 2176 7033 26 21 45] [ 92 82 94 5877 5879 34 45] [ 92 89 94 5877 5879 34 45] References ---------- .. [bfs] Edward Moore, "The shortest path through a maze", International Symposium on the Theory of Switching (1959), Harvard University Press .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with graphs." Numerische Mathematik, 1:269-271, 1959. .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html .. [bellman-ford] http://www.boost.org/libs/graph/doc/bellman_ford_shortest.html """ if dist_map is None or pred_map is None: dist_map, pred_map = shortest_distance(g, source, weights=weights, negative_weights=negative_weights, pred_map=True) if all_preds_map is None: all_preds_map = all_predecessors(g, dist_map, pred_map, weights, epsilon) path_iterator = \ libgraph_tool_topology.get_all_shortest_paths(g._Graph__graph, int(source), int(target), _prop("v", g, all_preds_map)) return path_iterator
[docs]def all_paths(g, source, target, cutoff=None): """Return an iterator over all paths from `source` to `target`. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. source : :class:`~graph_tool.Vertex` Source vertex of the search. target : :class:`~graph_tool.Vertex` Target vertex of the search. cutoff : `int` (optional, default: None) Maximum path length. Returns ------- path_iterator : iterator over a sequence of integers Iterator over sequences of vertices from `source` to `target` in the path. Notes ----- The algorithm uses a depth-first search to find all the paths. The total number of paths between any two vertices can be quite large, possibly scaling as :math:`O(V!)`. Examples -------- >>> g = gt.collection.data["football"] >>> for path in gt.all_paths(g, 13, 2, cutoff=2): ... print(path) [13 2] [13 15 2] [13 60 2] [13 64 2] [ 13 100 2] [ 13 106 2] """ if cutoff is None: cutoff = g.num_edges() + 1 visited = g.new_vp("bool", False) path_iterator = libgraph_tool_topology.get_all_paths(g._Graph__graph, int(source), int(target), cutoff, _prop("v", g, visited)) return path_iterator
[docs]def all_circuits(g, unique=False): """Return an iterator over all the cycles in a directed graph. Parameters ---------- g : :class:`~graph_tool.Graph` A directed graph to be used. unique : ``bool`` (optional, default: None) If ``True``, parallel edges and self-loops will be ignored. Returns ------- cycle_iterator : iterator over a sequence of integers Iterator over sequences of vertices that form a circuit. Notes ----- This algorithm [hawick-enumerating-2008]_ runs in worse time :math:`O[(V + E)(C + 1)]`, where :math:`C` is the number of circuits. Examples -------- .. testcode:: :hide: gt.seed_rng(42) >>> g = gt.random_graph(10, lambda: (1, 1)) >>> for c in gt.all_circuits(g): ... print(c) [0 4 7 1 8 2] [3 9 6 5] References ---------- .. [hawick-enumerating-2008] K.A. Hawick and H.A. James, "Enumerating Circuits and Loops in Graphs with Self-Arcs and Multiple-Arcs.", In Proceedings of FCS. 2008, 14-20, http://cssg.massey.ac.nz/cstn/013/cstn-013.html .. [hawick-bgl] http://www.boost.org/doc/libs/graph/doc/hawick_circuits.html """ if not g.is_directed(): raise ValueError("The graph must be directed.") circuits_iterator = libgraph_tool_topology.get_all_circuits(g._Graph__graph, unique) return circuits_iterator
[docs]def pseudo_diameter(g, source=None, weights=None): """ Compute the pseudo-diameter of the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. source : :class:`~graph_tool.Vertex` (optional, default: `None`) Source vertex of the search. If not supplied, the first vertex in the graph will be chosen. weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`) The edge weights. Returns ------- pseudo_diameter : int The pseudo-diameter of the graph. end_points : pair of :class:`~graph_tool.Vertex` The two vertices which correspond to the pseudo-diameter found. Notes ----- The pseudo-diameter is an approximate graph diameter. It is obtained by starting from a vertex `source`, and finds a vertex `target` that is farthest away from `source`. This process is repeated by treating `target` as the new starting vertex, and ends when the graph distance no longer increases. A vertex from the last level set that has the smallest degree is chosen as the final starting vertex u, and a traversal is done to see if the graph distance can be increased. This graph distance is taken to be the pseudo-diameter. The paths are computed with a breadth-first search (BFS) or Dijkstra's algorithm [dijkstra]_, if weights are given. The algorithm runs in :math:`O(V + E)` time, or :math:`O(V \log V)` if weights are given. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> from numpy.random import poisson >>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3))) >>> dist, ends = gt.pseudo_diameter(g) >>> print(dist) 10.0 >>> print(int(ends[0]), int(ends[1])) 0 11 References ---------- .. [pseudo-diameter] http://en.wikipedia.org/wiki/Distance_%28graph_theory%29 """ if source is None: source = g.vertex(0, use_index=False) dist, target = 0, source while True: new_source = target new_target, new_dist = libgraph_tool_topology.get_diam(g._Graph__graph, int(new_source), _prop("e", g, weights)) if new_dist > dist: target = new_target source = new_source dist = new_dist else: break return dist, (g.vertex(source), g.vertex(target))
[docs]def is_bipartite(g, partition=False, find_odd_cycle=False): """Test if the graph is bipartite. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. partition : bool (optional, default: ``False``) If ``True``, return the two partitions in case the graph is bipartite. find_odd_cycle : bool (optional, default: ``False``) If ``True``, return an odd cycle if the graph is not bipartite. Returns ------- is_bipartite : ``bool`` Whether or not the graph is bipartite. partition : :class:`~graph_tool.PropertyMap` (only if ``partition=True``) A vertex property map with the graph partitioning (or ``None``) if the graph is not bipartite. odd_cycle : list of vertices (only if ``find_odd_cycle=True``) A list of vertices corresponding to an odd cycle, or ``None`` if none is found. Notes ----- An undirected graph is bipartite if one can partition its set of vertices into two sets, such that all edges go from one set to the other. This algorithm runs in :math:`O(V + E)` time. Examples -------- >>> g = gt.lattice([10, 10]) >>> is_bi, part = gt.is_bipartite(g, partition=True) >>> print(is_bi) True >>> gt.graph_draw(g, vertex_fill_color=part, output_size=(300, 300), output="bipartite.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, vertex_fill_color=part, output_size=(300, 300), output="bipartite.png") .. figure:: bipartite.* :align: center Bipartition of a 2D lattice. References ---------- .. [boost-bipartite] http://www.boost.org/libs/graph/doc/is_bipartite.html """ if partition: part = g.new_vertex_property("bool") else: part = None g = GraphView(g, directed=False, skip_properties=True) cycle = [] is_bi = libgraph_tool_topology.is_bipartite(g._Graph__graph, _prop("v", g, part), find_odd_cycle, cycle) if not is_bi and part is not None: part.a = 0 if len(cycle) == 0: cycle = None else: cycle.append(cycle[0]) if find_odd_cycle: if partition: return is_bi, part, cycle else: return is_bi, cycle else: if partition: return is_bi, part else: return is_bi
[docs]def is_planar(g, embedding=False, kuratowski=False): """ Test if the graph is planar. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. embedding : bool (optional, default: False) If true, return a mapping from vertices to the clockwise order of out-edges in the planar embedding. kuratowski : bool (optional, default: False) If true, the minimal set of edges that form the obstructing Kuratowski subgraph will be returned as a property map, if the graph is not planar. Returns ------- is_planar : bool Whether or not the graph is planar. embedding : :class:`~graph_tool.PropertyMap` (only if `embedding=True`) A vertex property map with the out-edges indexes in clockwise order in the planar embedding, kuratowski : :class:`~graph_tool.PropertyMap` (only if `kuratowski=True`) An edge property map with the minimal set of edges that form the obstructing Kuratowski subgraph (if the value of kuratowski[e] is 1, the edge belongs to the set) Notes ----- A graph is planar if it can be drawn in two-dimensional space without any of its edges crossing. This algorithm performs the Boyer-Myrvold planarity testing [boyer-myrvold]_. See [boost-planarity]_ for more details. This algorithm runs in :math:`O(V)` time. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> from numpy.random import random >>> g = gt.triangulation(random((100,2)))[0] >>> p, embed_order = gt.is_planar(g, embedding=True) >>> print(p) True >>> print(list(embed_order[g.vertex(0)])) [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1] >>> g = gt.random_graph(100, lambda: 4, directed=False) >>> p, kur = gt.is_planar(g, kuratowski=True) >>> print(p) False >>> g.set_edge_filter(kur, True) >>> gt.graph_draw(g, output_size=(300, 300), output="kuratowski.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, output_size=(300, 300), output="kuratowski.png") .. figure:: kuratowski.* :align: center Obstructing Kuratowski subgraph of a random graph. References ---------- .. [boyer-myrvold] John M. Boyer and Wendy J. Myrvold, "On the Cutting Edge: Simplified O(n) Planarity by Edge Addition" Journal of Graph Algorithms and Applications, 8(2): 241-273, 2004. http://www.emis.ams.org/journals/JGAA/accepted/2004/BoyerMyrvold2004.8.3.pdf .. [boost-planarity] http://www.boost.org/libs/graph/doc/boyer_myrvold.html """ u = GraphView(g, directed=False) if embedding: embed = g.new_vertex_property("vector<int>") else: embed = None if kuratowski: kur = g.new_edge_property("bool") else: kur = None is_planar = libgraph_tool_topology.is_planar(u._Graph__graph, _prop("v", g, embed), _prop("e", g, kur)) ret = [is_planar] if embed is not None: ret.append(embed) if kur is not None: ret.append(kur) if len(ret) == 1: return ret[0] else: return tuple(ret)
[docs]def make_maximal_planar(g): """ Add edges to the graph to make it maximally planar. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. It must be a biconnected planar graph with at least 3 vertices. Notes ----- A graph is maximal planar if no additional edges can be added to it without creating a non-planar graph. By Euler's formula, a maximal planar graph with V > 2 vertices always has 3V - 6 edges and 2V - 4 faces. The input graph to make_maximal_planar() must be a biconnected planar graph with at least 3 vertices. This algorithm runs in :math:`O(V + E)` time. Examples -------- >>> g = gt.lattice([10, 10]) >>> gt.make_maximal_planar(g) >>> gt.is_planar(g) True >>> print(g.num_vertices(), g.num_edges()) 100 294 >>> pos = gt.planar_layout(g) >>> gt.graph_draw(g, pos, output_size=(300, 300), output="maximal_planar.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, pos, output_size=(300, 300), output="maximal_planar.png") .. figure:: maximal_planar.* :align: center A maximally planar graph. References ---------- .. [boost-planarity] http://www.boost.org/libs/graph/doc/make_maximal_planar.html """ g = GraphView(g, directed=False) libgraph_tool_topology.maximal_planar(g._Graph__graph)
[docs]def is_DAG(g): """ Return `True` if the graph is a directed acyclic graph (DAG). Notes ----- The time complexity is :math:`O(V + E)`. Examples -------- .. testcode:: :hide: import numpy.random numpy.random.seed(42) gt.seed_rng(42) >>> g = gt.random_graph(30, lambda: (3, 3)) >>> print(gt.is_DAG(g)) False >>> tree = gt.min_spanning_tree(g) >>> g.set_edge_filter(tree) >>> print(gt.is_DAG(g)) True References ---------- .. [DAG-wiki] http://en.wikipedia.org/wiki/Directed_acyclic_graph """ topological_order = Vector_int32_t() is_DAG = libgraph_tool_topology.\ topological_sort(g._Graph__graph, topological_order) return is_DAG
[docs]def max_cardinality_matching(g, heuristic=False, weight=None, minimize=True, match=None): r"""Find a maximum cardinality matching in the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. heuristic : bool (optional, default: `False`) If true, a random heuristic will be used, which runs in linear time. weight : :class:`~graph_tool.PropertyMap` (optional, default: `None`) If provided, the matching will minimize the edge weights (or maximize if ``minimize == False``). This option has no effect if ``heuristic == False``. minimize : bool (optional, default: `True`) If `True`, the matching will minimize the weights, otherwise they will be maximized. This option has no effect if ``heuristic == False``. match : :class:`~graph_tool.PropertyMap` (optional, default: `None`) Edge property map where the matching will be specified. Returns ------- match : :class:`~graph_tool.PropertyMap` Boolean edge property map where the matching is specified. Notes ----- A *matching* is a subset of the edges of a graph such that no two edges share a common vertex. A *maximum cardinality matching* has maximum size over all matchings in the graph. If the parameter ``weight`` is provided, as well as ``heuristic == True`` a matching with maximum cardinality *and* maximum (or minimum) weight is returned. If ``heuristic == True`` the algorithm does not necessarily return the maximum matching, instead the focus is to run on linear time. This algorithm runs in time :math:`O(EV\times\alpha(E,V))`, where :math:`\alpha(m,n)` is a slow growing function that is at most 4 for any feasible input. If `heuristic == True`, the algorithm runs in time :math:`O(V + E)`. For a more detailed description, see [boost-max-matching]_. Examples -------- .. testcode:: :hide: numpy.random.seed(43) gt.seed_rng(43) >>> g = gt.GraphView(gt.price_network(300), directed=False) >>> res = gt.max_cardinality_matching(g) >>> print(res[1]) True >>> w = res[0].copy("double") >>> w.a = 2 * w.a + 2 >>> gt.graph_draw(g, edge_color=res[0], edge_pen_width=w, vertex_fill_color="grey", ... output="max_card_match.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, edge_color=res[0], edge_pen_width=w, vertex_fill_color="grey", output="max_card_match.png") .. figure:: max_card_match.* :align: center Edges belonging to the matching are in yellow. References ---------- .. [boost-max-matching] http://www.boost.org/libs/graph/doc/maximum_matching.html .. [matching-heuristic] B. Hendrickson and R. Leland. "A Multilevel Algorithm for Partitioning Graphs." In S. Karin, editor, Proc. Supercomputing ’95, San Diego. ACM Press, New York, 1995, :doi:`10.1145/224170.224228` """ if match is None: match = g.new_edge_property("bool") _check_prop_scalar(match, "match") _check_prop_writable(match, "match") if weight is not None: _check_prop_scalar(weight, "weight") u = GraphView(g, directed=False) if not heuristic: check = libgraph_tool_flow.\ max_cardinality_matching(u._Graph__graph, _prop("e", u, match)) return match, check else: libgraph_tool_topology.\ random_matching(u._Graph__graph, _prop("e", u, weight), _prop("e", u, match), minimize, _get_rng()) return match
[docs]def max_independent_vertex_set(g, high_deg=False, mivs=None): r"""Find a maximal independent vertex set in the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. high_deg : bool (optional, default: `False`) If `True`, vertices with high degree will be included first in the set, otherwise they will be included last. mivs : :class:`~graph_tool.PropertyMap` (optional, default: `None`) Vertex property map where the vertex set will be specified. Returns ------- mivs : :class:`~graph_tool.PropertyMap` Boolean vertex property map where the set is specified. Notes ----- A maximal independent vertex set is an independent set such that adding any other vertex to the set forces the set to contain an edge between two vertices of the set. This implements the algorithm described in [mivs-luby]_, which runs in time :math:`O(V + E)`. Examples -------- .. testcode:: :hide: numpy.random.seed(43) gt.seed_rng(43) >>> g = gt.GraphView(gt.price_network(300), directed=False) >>> res = gt.max_independent_vertex_set(g) >>> gt.graph_draw(g, vertex_fill_color=res, output="mivs.pdf") <...> .. testcode:: :hide: gt.graph_draw(g, vertex_fill_color=res, output="mivs.png") .. figure:: mivs.* :align: center Vertices belonging to the set are in yellow. References ---------- .. [mivs-wikipedia] http://en.wikipedia.org/wiki/Independent_set_%28graph_theory%29 .. [mivs-luby] Luby, M., "A simple parallel algorithm for the maximal independent set problem", Proc. 17th Symposium on Theory of Computing, Association for Computing Machinery, pp. 1-10, (1985) :doi:`10.1145/22145.22146`. """ if mivs is None: mivs = g.new_vertex_property("bool") _check_prop_scalar(mivs, "mivs") _check_prop_writable(mivs, "mivs") u = GraphView(g, directed=False) libgraph_tool_topology.\ maximal_vertex_set(u._Graph__graph, _prop("v", u, mivs), high_deg, _get_rng()) mivs = g.own_property(mivs) return mivs
[docs]def edge_reciprocity(g): r"""Calculate the edge reciprocity of the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used edges. Returns ------- reciprocity : float The reciprocity value. Notes ----- The edge [reciprocity]_ is defined as :math:`E^\leftrightarrow/E`, where :math:`E^\leftrightarrow` and :math:`E` are the number of bidirectional and all edges in the graph, respectively. The algorithm runs with complexity :math:`O(E + V)`. Examples -------- >>> g = gt.Graph() >>> g.add_vertex(2) <...> >>> g.add_edge(g.vertex(0), g.vertex(1)) <...> >>> gt.edge_reciprocity(g) 0.0 >>> g.add_edge(g.vertex(1), g.vertex(0)) <...> >>> gt.edge_reciprocity(g) 1.0 >>> g = gt.collection.data["pgp-strong-2009"] >>> gt.edge_reciprocity(g) 0.692196963163... References ---------- .. [reciprocity] S. Wasserman and K. Faust, "Social Network Analysis". (Cambridge University Press, Cambridge, 1994) .. [lopez-reciprocity-2007] Gorka Zamora-López, Vinko Zlatić, Changsong Zhou, Hrvoje Štefančić, and Jürgen Kurths "Reciprocity of networks with degree correlations and arbitrary degree sequences", Phys. Rev. E 77, 016106 (2008) :doi:`10.1103/PhysRevE.77.016106`, :arxiv:`0706.3372` """ r = libgraph_tool_topology.reciprocity(g._Graph__graph) return r
[docs]def tsp_tour(g, src, weight=None): """Return a traveling salesman tour of the graph, which is guaranteed to be twice as long as the optimal tour in the worst case. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. The graph must be undirected. src : :class:`~graph_tool.Vertex` The source (and target) of the tour. weight : :class:`~graph_tool.PropertyMap` (optional, default: None) Edge weights. Returns ------- tour : :class:`numpy.ndarray` List of vertex indexes corresponding to the tour. Notes ----- The algorithm runs with :math:`O(E\log V)` complexity. Examples -------- >>> g = gt.lattice([10, 10]) >>> tour = gt.tsp_tour(g, g.vertex(0)) >>> print(tour) [ 0 1 2 11 12 21 22 31 32 41 42 51 52 61 62 71 72 81 82 83 73 63 53 43 33 23 13 3 4 5 6 7 8 9 19 29 39 49 59 69 79 89 14 24 34 44 54 64 74 84 91 92 93 94 95 85 75 65 55 45 35 25 15 16 17 18 27 28 37 38 47 48 57 58 67 68 77 78 87 88 97 98 99 26 36 46 56 66 76 86 96 10 20 30 40 50 60 70 80 90 0] References ---------- .. [tsp-bgl] http://www.boost.org/libs/graph/doc/metric_tsp_approx.html .. [tsp] http://en.wikipedia.org/wiki/Travelling_salesman_problem """ if g.is_directed(): raise ValueError("The graph must be undirected.") tour = libgraph_tool_topology.\ get_tsp(g._Graph__graph, int(src), _prop("e", g, weight)) return tour.a.copy()
[docs]def sequential_vertex_coloring(g, order=None, color=None): """Returns a vertex coloring of the graph. Parameters ---------- g : :class:`~graph_tool.Graph` Graph to be used. order : :class:`~graph_tool.PropertyMap` (optional, default: None) Order with which the vertices will be colored. color : :class:`~graph_tool.PropertyMap` (optional, default: None) Integer-valued vertex property map to store the colors. Returns ------- color : :class:`~graph_tool.PropertyMap` Integer-valued vertex property map with the vertex colors. Notes ----- The time complexity is :math:`O(V(d+k))`, where :math:`V` is the number of vertices, :math:`d` is the maximum degree of the vertices in the graph, and :math:`k` is the number of colors used. Examples -------- >>> g = gt.lattice([10, 10]) >>> colors = gt.sequential_vertex_coloring(g) >>> print(colors.a) [0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0] References ---------- .. [sgc-bgl] http://www.boost.org/libs/graph/doc/sequential_vertex_coloring.html .. [graph-coloring] http://en.wikipedia.org/wiki/Graph_coloring """ if order is None: order = g.vertex_index if color is None: color = g.new_vertex_property("int") libgraph_tool_topology.\ sequential_coloring(g._Graph__graph, _prop("v", g, order), _prop("v", g, color)) return color
from .. flow import libgraph_tool_flow