blockmodel.py 121 KB
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#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
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Tiago Peixoto committed
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# Copyright (C) 2006-2015 Tiago de Paula Peixoto <tiago@skewed.de>
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#
# 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/>.

from __future__ import division, absolute_import, print_function
import sys
if sys.version_info < (3,):
    range = xrange

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from .. import _degree, _prop, Graph, GraphView, libcore, _get_rng, PropertyMap
from .. stats import label_self_loops
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from .. spectral import adjacency
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import random
from numpy import *
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import numpy
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from scipy.optimize import fsolve, fminbound
import scipy.special
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from collections import defaultdict
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import copy
import heapq
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from .. dl_import import dl_import
dl_import("from . import libgraph_tool_community as libcommunity")

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__test__ = False
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def set_test(test):
    global __test__
    __test__ = test

def _bm_test():
    global __test__
    return __test__

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def get_block_graph(g, B, b, vcount, ecount):
    cg, br, vcount, ecount = condensation_graph(g, b,
                                                vweight=vcount,
                                                eweight=ecount,
                                                self_loops=True)[:4]
    cg.vp["count"] = vcount
    cg.ep["count"] = ecount
    cg = Graph(cg, vorder=br)

    cg.add_vertex(B - cg.num_vertices())
    return cg

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class BlockState(object):
    r"""This class encapsulates the block state of a given graph.

    This must be instantiated and used by functions such as :func:`mcmc_sweep`.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
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        Graph to be modelled.
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    eweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
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        Edge multiplicities (for multigraphs or block graphs).
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    vweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
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        Vertex multiplicities (for block graphs).
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    b : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Initial block labels on the vertices. If not supplied, it will be
        randomly sampled.
    B : ``int`` (optional, default: ``None``)
        Number of blocks. If not supplied it will be either obtained from the
        parameter ``b``, or set to the maximum possible value according to the
        minimum description length.
    clabel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
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        Constraint labels on the vertices. If supplied, vertices with different
        label values will not be clustered in the same group.
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    deg_corr : ``bool`` (optional, default: ``True``)
        If ``True``, the degree-corrected version of the blockmodel ensemble will
        be assumed, otherwise the traditional variant will be used.
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    max_BE : ``int`` (optional, default: ``1000``)
        If the number of blocks exceeds this number, a sparse representation of
        the block graph is used, which is slightly less efficient, but uses less
        memory,
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    """

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    _state_ref_count = 0

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    def __init__(self, g, eweight=None, vweight=None, b=None,
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                 B=None, clabel=None, deg_corr=True,
                 max_BE=1000, **kwargs):
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        BlockState._state_ref_count += 1

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        # initialize weights to unity, if necessary
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        if eweight is None:
            eweight = g.new_edge_property("int")
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            eweight.fa = 1
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        elif eweight.value_type() != "int32_t":
            eweight = eweight.copy(value_type="int32_t")
        if vweight is None:
            vweight = g.new_vertex_property("int")
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            vweight.fa = 1
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        elif vweight.value_type() != "int32_t":
            vweight = vweight.copy(value_type="int32_t")
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        self.eweight = g.own_property(eweight)
        self.vweight = g.own_property(vweight)

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        self.is_weighted = False
        if ((g.num_edges() > 0 and self.eweight.fa.max() > 1) or
            kwargs.get("force_weighted", False)):
            self.is_weighted = True
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        # configure the main graph and block model parameters
        self.g = g
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        self.E = int(self.eweight.fa.sum())
        self.N = int(self.vweight.fa.sum())
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        self.deg_corr = deg_corr

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        # ensure we have at most as many blocks as nodes
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        if B is not None and b is None:
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            B = min(B, self.g.num_vertices())

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        if b is None:
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            # create a random partition into B blocks.
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            if B is None:
                B = get_max_B(self.N, self.E, directed=g.is_directed())
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            B = min(B, self.g.num_vertices())
            ba = random.randint(0, B, self.g.num_vertices())
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            ba[:B] = arange(B)        # avoid empty blocks
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            if B < self.g.num_vertices():
                random.shuffle(ba)
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            b = g.new_vertex_property("int")
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            b.fa = ba
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            self.b = b
        else:
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            # if a partition is available, we will incorporate it.
            if isinstance(b, numpy.ndarray):
                self.b = g.new_vertex_property("int")
                self.b.fa = b
            else:
                self.b = b = g.own_property(b.copy(value_type="int"))
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            if B is None:
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                B = int(self.b.fa.max()) + 1

        # if B > self.N:
        #     raise ValueError("B > N!")
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        if self.b.fa.max() >= B:
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            raise ValueError("Maximum value of b is larger or equal to B! (%d vs %d)" % (self.b.fa.max(), B))
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        # Construct block-graph
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        self.bg = get_block_graph(g, B, self.b, self.vweight, self.eweight)
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        self.bg.set_fast_edge_removal()

        self.mrs = self.bg.ep["count"]
        self.wr = self.bg.vp["count"]
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        del self.bg.ep["count"]
        del self.bg.vp["count"]

        self.mrp = self.bg.degree_property_map("out", weight=self.mrs)

        if g.is_directed():
            self.mrm = self.bg.degree_property_map("in", weight=self.mrs)
        else:
            self.mrm = self.mrp
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        self.vertices = libcommunity.get_vector(B)
        self.vertices.a = arange(B)
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        self.B = B
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        if clabel is not None:
            if isinstance(clabel, PropertyMap):
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                self.clabel = self.g.own_property(clabel.copy("int"))
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            else:
                self.clabel = self.g.new_vertex_property("int")
                self.clabel.a = clabel
        else:
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            self.clabel = self.g.new_vertex_property("int")

        self.emat = None
        if max_BE is None:
            max_BE = 1000
        self.max_BE = max_BE

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        self.overlap = False
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        self.ignore_degrees = kwargs.get("ignore_degrees", None)
        if self.ignore_degrees is None:
            self.ignore_degrees = g.new_vertex_property("bool", False)
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        # used by mcmc_sweep()
        self.egroups = None
        self.nsampler = None
        self.sweep_vertices = None
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        self.overlap_stats = libcommunity.overlap_stats()
        self.partition_stats = libcommunity.partition_stats()
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        self.edges_dl = False
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    def __repr__(self):
        return "<BlockState object with %d blocks,%s for graph %s, at 0x%x>" % \
            (self.B, " degree corrected," if self.deg_corr else "", str(self.g),
             id(self))


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    def __init_partition_stats(self, empty=True, edges_dl=False):
        self.edges_dl = edges_dl
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        if not empty:
            self.partition_stats = libcommunity.init_partition_stats(self.g._Graph__graph,
                                                                     _prop("v", self.g, self.b),
                                                                     _prop("e", self.g, self.eweight),
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                                                                     self.N, self.B,
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                                                                     edges_dl,
                                                                     _prop("v", self.g, self.ignore_degrees))
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        else:
            self.partition_stats = libcommunity.partition_stats()



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    def copy(self, g=None, eweight=None, vweight=None, b=None, B=None,
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             deg_corr=None, clabel=None, overlap=False, **kwargs):
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        r"""Copies the block state. The parameters override the state properties, and
         have the same meaning as in the constructor. If ``overlap=True`` an
         instance of :class:`~graph_tool.community.OverlapBlockState` is
         returned."""

        if not overlap:
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            state = BlockState(self.g if g is None else g,
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                               eweight=self.eweight if eweight is None else eweight,
                               vweight=self.vweight if vweight is None else vweight,
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                               b=self.b.copy() if b is None else b,
                               B=(self.B if b is None else None) if B is None else B,
                               clabel=self.clabel if clabel is None else clabel,
                               deg_corr=self.deg_corr if deg_corr is None else deg_corr,
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                               max_BE=self.max_BE,
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                               ignore_degrees=kwargs.pop("ignore_degrees", self.ignore_degrees),
                               **kwargs)
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        else:
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            state = OverlapBlockState(self.g if g is None else g,
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                                      b=b if b is not None else self.b,
                                      B=(self.B if b is None else None) if B is None else B,
                                      clabel=self.clabel if clabel is None else clabel,
                                      deg_corr=self.deg_corr if deg_corr is None else deg_corr,
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                                      max_BE=self.max_BE, **kwargs)
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        if not state.__check_clabel():
            b = state.b.a + state.clabel.a * state.B
            continuous_map(b)
            state = state.copy(b=b)

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            if _bm_test():
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                assert state.__check_clabel()

        return state


    def __getstate__(self):
        state = dict(g=self.g,
                     eweight=self.eweight,
                     vweight=self.vweight,
                     b=self.b,
                     B=self.B,
                     clabel=self.clabel,
                     deg_corr=self.deg_corr,
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                     max_BE=self.max_BE,
                     ignore_degrees=self.ignore_degrees)
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        return state

    def __setstate__(self, state):
        self.__init__(**state)
        return state

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    def get_block_state(self, b=None, vweight=False, deg_corr=False,
                        overlap=False, **kwargs):
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        r"""Returns a :class:`~graph_tool.community.BlockState`` corresponding to the
        block graph. The parameters have the same meaning as the in the constructor."""


        state = BlockState(self.bg, eweight=self.mrs,
                           vweight=self.wr if vweight else None,
                           b=self.bg.vertex_index.copy("int") if b is None else b,
                           clabel=self.get_bclabel(),
                           deg_corr=deg_corr,
                           max_BE=self.max_BE)
        if overlap:
            state = state.copy(overlap=True)
        n_map = self.b.copy()
        return state, n_map

    def get_bclabel(self):
        r"""Returns a :class:`~graph_tool.PropertyMap`` corresponding to constraint
        labels for the block graph."""

        bclabel = self.bg.new_vertex_property("int")
        reverse_map(self.b, bclabel)
        pmap(bclabel, self.clabel)
        return bclabel

    def __check_clabel(self):
        b = self.b.a + self.clabel.a * self.B
        continuous_map(b)
        b2 = self.b.copy()
        continuous_map(b2.a)
        return (b == b2.a).all()

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    def __get_emat(self):
        if self.emat is None:
            self.__regen_emat()
        return self.emat
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    def __regen_emat(self):
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        if self.B <= self.max_BE:
            self.emat = libcommunity.create_emat(self.bg._Graph__graph)
        else:
            self.emat = libcommunity.create_ehash(self.bg._Graph__graph)
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    def __build_egroups(self, empty=False):
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        self.esrcpos = self.g.new_edge_property("int")
        self.etgtpos = self.g.new_edge_property("int")
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        self.egroups = libcommunity.build_egroups(self.g._Graph__graph,
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                                                  self.bg._Graph__graph,
                                                  _prop("v", self.g, self.b),
                                                  _prop("e", self.g, self.eweight),
                                                  _prop("e", self.g, self.esrcpos),
                                                  _prop("e", self.g, self.etgtpos),
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                                                  self.is_weighted, empty)
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    def __build_nsampler(self, empty=False):
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        self.nsampler = libcommunity.init_neighbour_sampler(self.g._Graph__graph,
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                                                            _prop("e", self.g, self.eweight),
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                                                            True, empty)
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    def __cleanup_bg(self):
        emask = self.bg.new_edge_property("bool")
        emask.a = self.mrs.a[:len(emask.a)] > 0
        self.bg.set_edge_filter(emask)
        self.bg.purge_edges()
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        self.emat = None
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    def get_blocks(self):
        r"""Returns the property map which contains the block labels for each vertex."""
        return self.b

    def get_bg(self):
        r"""Returns the block graph."""
        return self.bg

    def get_ers(self):
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        r"""Returns the edge property map of the block graph which contains the :math:`e_{rs}` matrix entries.
        For undirected graphs, the diagonal values (self-loops) contain :math:`e_{rr}/2`."""
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        return self.mrs

    def get_er(self):
        r"""Returns the vertex property map of the block graph which contains the number
        :math:`e_r` of half-edges incident on block :math:`r`. If the graph is
        directed, a pair of property maps is returned, with the number of
        out-edges :math:`e^+_r` and in-edges :math:`e^-_r`, respectively."""
        if self.bg.is_directed():
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            return self.mrp, self.mrm
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        else:
            return self.mrp

    def get_nr(self):
        r"""Returns the vertex property map of the block graph which contains the block sizes :math:`n_r`."""
        return self.wr

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    def entropy(self, complete=True, dl=False, partition_dl=True,
                degree_dl=True, edges_dl=True, dense=False, multigraph=True,
                norm=False, dl_ent=False, **kwargs):
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        r"""Calculate the entropy associated with the current block partition.
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        Parameters
        ----------
        complete : ``bool`` (optional, default: ``False``)
            If ``True``, the complete entropy will be returned, including constant
            terms not relevant to the block partition.
        dl : ``bool`` (optional, default: ``False``)
            If ``True``, the full description length will be returned.
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        partition_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the partition description length
            will be considered.
        edges_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the edge matrix description length
            will be considered.
        degree_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the degree sequence description
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            length will be considered.
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        dense : ``bool`` (optional, default: ``False``)
            If ``True``, the "dense" variant of the entropy will be computed.
        multigraph : ``bool`` (optional, default: ``False``)
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            If ``True``, the multigraph entropy will be used.
        norm : ``bool`` (optional, default: ``True``)
            If ``True``, the entropy will be "normalized" by dividing by the
            number of edges.
        dl_ent : ``bool`` (optional, default: ``False``)
            If ``True``, the description length of the degree sequence will be
            approximated by its entropy.
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        Notes
        -----
        For the traditional blockmodel (``deg_corr == False``), the entropy is
        given by

        .. math::

          \mathcal{S}_t &\cong E - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
          \mathcal{S}^d_t &\cong E - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

        for undirected and directed graphs, respectively, where :math:`e_{rs}`
        is the number of edges from block :math:`r` to :math:`s` (or the number
        of half-edges for the undirected case when :math:`r=s`), and :math:`n_r`
        is the number of vertices in block :math:`r` .

        For the degree-corrected variant with "hard" degree constraints the
        equivalent expressions are

        .. math::

            \mathcal{S}_c &\cong -E -\sum_kN_k\ln k! - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
            \mathcal{S}^d_c &\cong -E -\sum_{k^+}N_{k^+}\ln k^+!  -\sum_{k^-}N_{k^-}\ln k^-! - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

        where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
        block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
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        \sum_se_{sr}` are the numbers of out- and in-edges adjacent to block
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        :math:`r`, respectively.

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        If ``dense == False`` and ``multigraph == True``, the entropy used will
        be of the "Poisson" model, with the additional term:

        .. math::

            {\mathcal{S}_{cm}^{(d)}} = \mathcal{S}_c^{(d)} + \sum_{i>j} \ln A_{ij}! + \sum_i \ln A_{ii}!!


        If ``dl == True``, the description length :math:`\mathcal{L}_t` of the
        model will be returned as well, as described in
        :func:`model_entropy`. Note that for the degree-corrected version the
        description length is

        .. math::

            \mathcal{L}_c = \mathcal{L}_t + \sum_r\min\left(\mathcal{L}^{(1)}_r, \mathcal{L}^{(2)}_r\right),

        with

        .. math::

              \mathcal{L}^{(1)}_r &= \ln{\left(\!\!{n_r \choose e_r}\!\!\right)}, \\
              \mathcal{L}^{(2)}_r &= \ln\Xi_r + \ln n_r! - \sum_k \ln n^r_k!,

        and :math:`\ln\Xi_r \simeq 2\sqrt{\zeta(2)e_r}`, where :math:`\zeta(x)`
        is the `Riemann zeta function
        <https://en.wikipedia.org/wiki/Riemann_zeta_function>`_, and
        :math:`n^r_k` is the number of nodes in block :math:`r` with degree
        :math:`k`. For directed graphs we have instead :math:`k \to (k^-, k^+)`,
        and :math:`\ln\Xi_r \to \ln\Xi^+_r + \ln\Xi^-_r` with :math:`\ln\Xi_r^+
        \simeq 2\sqrt{\zeta(2)e^+_r}` and :math:`\ln\Xi_r^- \simeq
        2\sqrt{\zeta(2)e^-_r}`.

        If ``dl_ent=True`` is passed, this will be approximated instead by
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        .. math::

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            \mathcal{L}_c \simeq \mathcal{L}_t - \sum_rn_r\sum_kp^r_k\ln p^r_k,
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        where :math:`p^r_k = n^r_k / n_r`.
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        If the "dense" entropies are requested (``dense=True``), they will be
        computed as
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        .. math::

            \mathcal{S}_t  &= \sum_{r>s} \ln{\textstyle {n_rn_s \choose e_{rs}}} + \sum_r \ln{\textstyle {{n_r\choose 2}\choose e_{rr}/2}}\\
            \mathcal{S}^d_t  &= \sum_{rs} \ln{\textstyle {n_rn_s \choose e_{rs}}},

        for simple graphs, and

        .. math::

            \mathcal{S}_m  &= \sum_{r>s} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)} + \sum_r \ln{\textstyle \left(\!\!{\left(\!{n_r\choose 2}\!\right)\choose e_{rr}/2}\!\!\right)}\\
            \mathcal{S}^d_m  &= \sum_{rs} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)},

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        for multigraphs (i.e. ``multigraph == True``). A dense entropy for the
        degree-corrected model is not available, and if requested will return a
        :exc:`NotImplementedError`.
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        If ``complete == False`` constants in the above equations which do not
        depend on the partition of the nodes will be omitted.
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        Note that in all cases if ``norm==True`` the value returned corresponds
        to the entropy `per edge`, i.e. :math:`(\mathcal{S}_{t/c}\; [\,+\,\mathcal{L}_{t/c}])/ E`.
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        """

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        xi_fast = kwargs.get("xi_fast", False)
        dl_deg_alt = kwargs.get("dl_deg_alt", True)

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        E = self.E
        N = self.N

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        if dense:
            if self.deg_corr:
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                raise NotImplementedError('A degree-corrected "dense" entropy is not yet implemented')
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            S = libcommunity.entropy_dense(self.bg._Graph__graph,
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                                            _prop("e", self.bg, self.mrs),
                                            _prop("v", self.bg, self.wr),
                                            multigraph)
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        else:
            S = libcommunity.entropy(self.bg._Graph__graph,
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                                      _prop("e", self.bg, self.mrs),
                                      _prop("v", self.bg, self.mrp),
                                      _prop("v", self.bg, self.mrm),
                                      _prop("v", self.bg, self.wr),
                                      self.deg_corr)
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            if _bm_test():
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                assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                random=random, dl=dl,
                                                                                                partition_dl=partition_dl,
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                                                                                                edges_dl=edges_dl,
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                                                                                                dense=dense, multigraph=multigraph,
                                                                                                norm=norm)))
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            if self.deg_corr:
                S -= E
            else:
                S += E

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            if complete:
                if self.deg_corr:
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                    S += libcommunity.deg_entropy_term(self.g._Graph__graph,
                                                       libcore.any(),
                                                       self.overlap_stats,
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                                                       self.N,
                                                       _prop("e", self.g, self.eweight),
                                                       _prop("v", self.g, self.ignore_degrees))
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                if multigraph:
                    S += libcommunity.entropy_parallel(self.g._Graph__graph,
                                                       _prop("e", self.g, self.eweight))

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                if _bm_test():
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                    assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                    random=random, dl=dl,
                                                                                                    partition_dl=partition_dl,
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                                                                                                    edges_dl=edges_dl,
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                                                                                                    dense=dense, multigraph=multigraph,
                                                                                                    norm=norm)))
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        if dl:
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            if partition_dl:
                if self.partition_stats.is_enabled():
                    S += self.partition_stats.get_partition_dl()
                else:
                    self.__init_partition_stats(empty=False)
                    S += self.partition_stats.get_partition_dl()
                    self.__init_partition_stats(empty=True)

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                if _bm_test():
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                    assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                    random=random, dl=dl,
                                                                                                    partition_dl=partition_dl,
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                                                                                                    edges_dl=edges_dl,
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                                                                                                    dense=dense, multigraph=multigraph,
                                                                                                    norm=norm)))
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            if edges_dl:
                actual_B = (self.wr.a > 0).sum()
                S += model_entropy(actual_B, N, E, directed=self.g.is_directed(), nr=False)
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            if self.deg_corr and degree_dl:
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                if self.partition_stats.is_enabled():
                    S_seq = self.partition_stats.get_deg_dl(dl_ent, dl_deg_alt, xi_fast)
                else:
                    self.__init_partition_stats(empty=False)
                    S_seq = self.partition_stats.get_deg_dl(dl_ent, dl_deg_alt, xi_fast)
                    self.__init_partition_stats(empty=True)
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                S += S_seq
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                if _bm_test():
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                    assert not isnan(S_seq) and not isinf(S_seq), "invalid entropy %g (%s) " % (S_seq, str(dict(complete=complete,
                                                                                                                random=random, dl=dl,
                                                                                                                partition_dl=partition_dl,
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                                                                                                                edges_dl=edges_dl,
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                                                                                                                dense=dense, multigraph=multigraph,
                                                                                                                norm=norm)))

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        if _bm_test():
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            assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                            random=random, dl=dl,
                                                                                            partition_dl=partition_dl,
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                                                                                            edges_dl=edges_dl,
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                                                                                            dense=dense, multigraph=multigraph,
                                                                                            norm=norm)))

        if norm:
            return S / E
        else:
            return S
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    def get_matrix(self):
        r"""Returns the block matrix (as a sparse :class:`~scipy.sparse.csr_matrix`),
        which contains the number of edges between each block pair.
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        .. warning::

           This corresponds to the adjacency matrix of the block graph, which by
           convention includes twice the amount of edges in the diagonal entries
           if the graph is undirected.

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        Examples
        --------

        .. testsetup:: get_matrix

           gt.seed_rng(42)
           np.random.seed(42)
           from pylab import *

        .. doctest:: get_matrix

           >>> g = gt.collection.data["polbooks"]
           >>> state = gt.BlockState(g, B=5, deg_corr=True)
           >>> for i in range(1000):
           ...     ds, nmoves = gt.mcmc_sweep(state)
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           >>> m = state.get_matrix()
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           >>> figure()
           <...>
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           >>> matshow(m.todense())
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           <...>
           >>> savefig("bloc_mat.pdf")

        .. testcleanup:: get_matrix

           savefig("bloc_mat.png")

        .. figure:: bloc_mat.*
           :align: center

           A  5x5 block matrix.

       """
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        return adjacency(self.bg, weight=self.mrs)
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def model_entropy(B, N, E, directed=False, nr=None):
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    r"""Computes the amount of information necessary for the parameters of the traditional blockmodel ensemble, for ``B`` blocks, ``N`` vertices, ``E`` edges, and either a directed or undirected graph.

    A traditional blockmodel is defined as a set of :math:`N` vertices which can
    belong to one of :math:`B` blocks, and the matrix :math:`e_{rs}` describes
    the number of edges from block :math:`r` to :math:`s` (or twice that number
    if :math:`r=s` and the graph is undirected).

    For an undirected graph, the number of distinct :math:`e_{rs}` matrices is given by,

    .. math::

       \Omega_m = \left(\!\!{\left(\!{B \choose 2}\!\right) \choose E}\!\!\right)

    and for a directed graph,

    .. math::
       \Omega_m = \left(\!\!{B^2 \choose E}\!\!\right)


    where :math:`\left(\!{n \choose k}\!\right) = {n+k-1\choose k}` is the
    number of :math:`k` combinations with repetitions from a set of size :math:`n`.

    The total information necessary to describe the model is then,

    .. math::

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       \mathcal{L}_t = \ln\Omega_m + \ln\left(\!\!{B \choose N}\!\!\right) + \ln N! - \sum_r \ln n_r!,

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    where the remaining term is the information necessary to describe the
    block partition, where :math:`n_r` is the number of nodes in block :math:`r`.
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    If ``nr`` is ``None``, it is assumed :math:`n_r=N/B`.

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    References
    ----------

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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    .. [peixoto-hierarchical-2014] Tiago P. Peixoto, "Hierarchical block structures and high-resolution
       model selection in large networks ", Phys. Rev. X 4, 011047 (2014), :doi:`10.1103/PhysRevX.4.011047`,
       :arxiv:`1310.4377`.
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    """

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    if directed:
        x = (B * B);
    else:
        x = (B * (B + 1)) / 2;
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    if nr is False:
        L = lbinom(x + E - 1, E)
    else:
        L = lbinom(x + E - 1, E) + partition_entropy(B, N, nr)
    return L
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def lbinom(n, k):
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    return scipy.special.gammaln(float(n + 1)) - scipy.special.gammaln(float(n - k + 1)) - scipy.special.gammaln(float(k + 1))
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def lbinom_careful(n, k):
    return libcommunity.lbinom_careful(n, k)

def lbinom_fast(n, k):
    return libcommunity.lbinom_fast(n, k)

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def partition_entropy(B, N, nr=None):
    if nr is None:
        S = N * log(B) + log1p(-(1 - 1./B) ** N)
    else:
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        S = lbinom(B + N - 1, N) + scipy.special.gammaln(N + 1) - scipy.special.gammaln(nr + 1).sum()
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    return S
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def get_max_B(N, E, directed=False):
    r"""Return the maximum detectable number of blocks, obtained by minimizing:

    .. math::

        \mathcal{L}_t(B, N, E) - E\ln B

    where :math:`\mathcal{L}_t(B, N, E)` is the information necessary to
    describe a traditional blockmodel with `B` blocks, `N` nodes and `E`
    edges (see :func:`model_entropy`).

    Examples
    --------

    >>> gt.get_max_B(N=1e6, E=5e6)
    1572

    References
    ----------
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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    """

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    def Sdl(B, S, N, E, directed=False):
        return S + model_entropy(B, N, E, directed) / E

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    B = fminbound(lambda B: Sdl(B, -log(B), N, E, directed), 1, E,
                  xtol=1e-6, maxfun=1500, disp=0)
    if isnan(B):
        B = 1
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    return min(N, max(int(ceil(B)), 2))
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def get_akc(B, I, N=numpy.inf, directed=False):
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    r"""Return the minimum value of the average degree of the network, so that some block structure with :math:`B` blocks can be detected, according to the minimum description length criterion.
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    This is obtained by solving

    .. math::

       \Sigma_b = \mathcal{L}_t(B, N, E) - E\mathcal{I}_{t/c} = 0,

    where :math:`\mathcal{L}_{t}` is the necessary information to describe the
    blockmodel parameters (see :func:`model_entropy`), and
    :math:`\mathcal{I}_{t/c}` characterizes the planted block structure, and is
    given by

    .. math::

        \mathcal{I}_t &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{w_rw_s}\right),\\
        \mathcal{I}_c &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{m_rm_s}\right),

    where :math:`m_{rs} = e_{rs}/2E` (or :math:`m_{rs} = e_{rs}/E` for directed
    graphs) and :math:`w_r=n_r/N`. We note that :math:`\mathcal{I}_{t/c}\in[0,
    \ln B]`. In the case where :math:`E \gg B^2`, this simplifies to

    .. math::

       \left<k\right>_c &= \frac{2\ln B}{\mathcal{I}_{t/c}},\\
       \left<k^{-/+}\right>_c &= \frac{\ln B}{\mathcal{I}_{t/c}},

    for undirected and directed graphs, respectively. This limit is assumed if
    ``N == inf``.

    Examples
    --------

    >>> gt.get_akc(10, log(10) / 100, N=100)
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    2.414413200430159
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    References
    ----------
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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    """
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    if N != numpy.inf:
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        if directed:
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            get_dl = lambda ak: model_entropy(B, N, N * ak, directed) / N * ak - N * ak * I
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        else:
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            get_dl = lambda ak: model_entropy(B, N, N * ak / 2., directed) * 2 / (N * ak)  - N * ak * I / 2.
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        ak = fsolve(lambda ak: get_dl(ak), 10)
        ak = float(ak)
    else:
        ak = 2 * log(B) / S
        if directed:
            ak /= 2
    return ak

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def mcmc_sweep(state, beta=1., c=1., niter=1, dl=False, dense=False,
               multigraph=False, node_coherent=False, confine_layers=False,
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               sequential=True, parallel=False, vertices=None,
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               target_blocks=None, verbose=False, **kwargs):
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    r"""Performs a Markov chain Monte Carlo sweep on the network, to sample the block partition according to a probability :math:`\propto e^{-\beta \mathcal{S}_{t/c}}`, where :math:`\mathcal{S}_{t/c}` is the blockmodel entropy.
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    Parameters
    ----------
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    state : :class:`~graph_tool.community.BlockState`, :class:`~graph_tool.community.OverlapBlockState` or :class:`~graph_tool.community.CovariateBlockState`
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        The block state.
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    beta : ``float`` (optional, default: `1.0`)
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        The inverse temperature parameter :math:`\beta`.
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    c : ``float`` (optional, default: ``1.0``)
        This parameter specifies how often fully random moves are attempted,
        instead of more likely moves based on the inferred block partition.
        For ``c == 0``, no fully random moves are attempted, and for ``c == inf``
        they are always attempted.
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    niter : ``int`` (optional, default: ``1``)
        Number of sweeps to perform.
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    dl : ``bool`` (optional, default: ``False``)
        If ``True``, the change in the whole description length will be
        considered after each vertex move, not only the entropy.
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    dense : ``bool`` (optional, default: ``False``)
        If ``True``, the "dense" variant of the entropy will be computed.
    multigraph : ``bool`` (optional, default: ``False``)
        If ``True``, the multigraph entropy will be used. Only has an effect
        if ``dense == True``.
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    node_coherent : ``bool`` (optional, default: ``False``)
        If ``True``, and if the ``state`` is an instance of
        :class:`~graph_tool.community.OverlapBlockState`, then all half-edges
        incident on the same node are moved simultaneously.
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    confine_layers : ``bool`` (optional, default: ``False``)
        If ``True``, and if the ``state`` is an instance of
        :class:`~graph_tool.community.CovariateBlockState`, with an
        *overlapping* partition, the half edges will only be moved in such a way
         that inside each layer the group membership remains non-overlapping.
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    sequential : ``bool`` (optional, default: ``True``)
        If ``True``, the move attempts on the vertices are done in sequential
        random order. Otherwise a total of `N` moves attempts are made, where
        `N` is the number of vertices, where each vertex can be selected with
        equal probability.
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    parallel : ``bool`` (optional, default: ``False``)
        If ``True``, the updates are performed in parallel (multiple
        threads).

        .. warning::

            If this is used, the Markov Chain is not guaranteed to be sampled with
            the correct probabilities. This is better used in conjunction with
            ``beta=float('inf')``, where this is not an issue.

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    vertices : ``list of ints`` (optional, default: ``None``)
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        A list of vertices which will be attempted to be moved. If ``None``, all
        vertices will be attempted.
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    target_blocks : ``list of ints`` (optional, default: ``None``)
        A list of groups to which the corresponding vertices will to be forcibly
        moved. If ``None``, the standard MCMC rules will be applied.
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    verbose : ``bool`` (optional, default: ``False``)
        If ``True``, verbose information is displayed.

    Returns
    -------

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    dS : ``float``
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       The entropy difference (in nats) after the sweeps.
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    nmoves : ``int``
       The number of accepted block membership moves.


    Notes
    -----

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    This algorithm performs a Markov chain Monte Carlo sweep on the network,
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    where the block memberships are randomly moved, and either accepted or
    rejected, so that after sufficiently many sweeps the partitions are sampled
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    with probability proportional to :math:`e^{-\beta\mathcal{S}_{t/c}}`, where
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    :math:`\mathcal{S}_{t/c}` is the blockmodel entropy, given by

    .. math::

      \mathcal{S}_t &\cong - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
      \mathcal{S}^d_t &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

    for undirected and directed traditional blockmodels (``deg_corr == False``),
    respectively, where :math:`e_{rs}` is the number of edges from block
    :math:`r` to :math:`s` (or the number of half-edges for the undirected case
    when :math:`r=s`), and :math:`n_r` is the number of vertices in block
    :math:`r`, and constant terms which are independent of the block partition
    were dropped (see :meth:`BlockState.entropy` for the complete entropy). For
    the degree-corrected variant with "hard" degree constraints the equivalent
    expressions are

    .. math::

       \mathcal{S}_c &\cong  - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
       \mathcal{S}^d_c &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

    where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
    block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
    \sum_se_{sr}` are the number of out- and in-edges adjacent to block
    :math:`r`, respectively.

    The Monte Carlo algorithm employed attempts to improve the mixing time of
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    the Markov chain by proposing membership moves :math:`r\to s` with
    probability :math:`p(r\to s|t) \propto e_{ts} + c`, where :math:`t` is the
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    block label of a random neighbour of the vertex being moved. See
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    [peixoto-efficient-2014]_ for more details.
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    This algorithm has a complexity of :math:`O(E)`, where :math:`E` is the
    number of edges in the network.

    Examples
    --------
    .. testsetup:: mcmc

       gt.seed_rng(42)
       np.random.seed(42)

    .. doctest:: mcmc

       >>> g = gt.collection.data["polbooks"]
       >>> state = gt.BlockState(g, B=3, deg_corr=True)
       >>> pv = None
       >>> for i in range(1000):        # remove part of the transient
       ...     ds, nmoves = gt.mcmc_sweep(state)
       >>> for i in range(1000):
       ...     ds, nmoves = gt.mcmc_sweep(state)
       ...     pv = gt.collect_vertex_marginals(state, pv)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.pdf")
       <...>

    .. testcleanup:: mcmc

       gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.png")

    .. figure:: polbooks_blocks_soft.*
       :align: center

       "Soft" block partition of a political books network with :math:`B=3`.

     References
    ----------

    .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey,
       Samuel Leinhardt, "Stochastic blockmodels: First steps",
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       Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.,
       :doi:`10.1016/0378-8733(83)90021-7`
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    .. [faust-blockmodels-1992] Katherine Faust, and Stanley
       Wasserman. "Blockmodels: Interpretation and Evaluation." Social Networks
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       14, no. 1-2 (1992): 5-61. :doi:`10.1016/0378-8733(92)90013-W`
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    .. [karrer-stochastic-2011] Brian Karrer, and M. E. J. Newman. "Stochastic
       Blockmodels and Community Structure in Networks." Physical Review E 83,
       no. 1 (2011): 016107. :doi:`10.1103/PhysRevE.83.016107`.
    .. [peixoto-entropy-2012] Tiago P. Peixoto "Entropy of Stochastic Blockmodel
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       Ensembles." Physical Review E 85, no. 5 (2012): 056122.
       :doi:`10.1103/PhysRevE.85.056122`, :arxiv:`1112.6028`.
    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module
       inference in large networks", Phys. Rev. Lett. 110, 148701 (2013),
       :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    .. [peixoto-efficient-2014] Tiago P. Peixoto, "Efficient Monte Carlo and greedy
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       heuristic for the inference of stochastic block models", Phys. Rev. E 89,
       012804 (2014), :doi:`10.1103/PhysRevE.89.012804`, :arxiv:`1310.4378`.
    .. [peixoto-model-2015] Tiago P. Peixoto, "Model selection and hypothesis
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       testing for large-scale network models with overlapping groups",
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       Phys. Rev. X 5, 011033 (2015), :doi:`10.1103/PhysRevX.5.011033`,
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    """

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    nmerges = kwargs.get("nmerges", 0)
    merge_map = kwargs.get("merge_map", None)
    coherent_merge = kwargs.get("coherent_merge", False)
    edges_dl = kwargs.get("edges_dl", False)

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    if state.B == 1:
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        return 0., 0

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    if vertices is not None:
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        temp = libcommunity.get_vector(len(vertices))
        temp.a = vertices
        vertices = temp
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        state.sweep_vertices = vertices
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    elif (state.sweep_vertices is None or
          len(state.sweep_vertices.a) < state.g.num_vertices()):
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        vertices = libcommunity.get_vector(state.g.num_vertices())
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