Update documentation

.gitlab-ci.yml

@@ -19,47 +19,3 @@ job_clang_amd64:
     - cd doc; python3 /usr/bin/sphinx-build -b doctest . build *.rst
   tags:
     - amd64
-
-# job_gcc_py2_amd64:
-#   script:
-#     - ./autogen.sh
-#     - ./configure CXX="ccache g++" PYTHON=python2 --prefix=$PWD/install --with-python-module-path=$PWD/install/site-packages
-#     - CCACHE_BASEDIR=$PWD make $MAKEOPTS
-#     - make install
-#     - export PYTHONPATH=$PWD/install/site-packages
-#     - cd doc; python2 /usr/bin/sphinx-build2 -b doctest . build *.rst demos/inference/inference.rst
-#   tags:
-#     - amd64
-
-# job_clang_py2_amd64:
-#   script:
-#     - ./autogen.sh
-#     - ./configure CXX="ccache clang++" PYTHON=python2 --prefix=$PWD/install --with-python-module-path=$PWD/install/site-packages
-#     - CCACHE_BASEDIR=$PWD make $MAKEOPTS
-#     - make install
-#     - export PYTHONPATH=$PWD/install/site-packages
-#     - cd doc; python2 /usr/bin/sphinx-build2 -b doctest . build *.rst
-#   tags:
-#     - amd64
-
-job_gcc_amd64_nosh:
-  script:
-    - ./autogen.sh
-    - ./configure CXX="ccache g++" PYTHON=python3 --prefix=$PWD/install --with-python-module-path=$PWD/install/site-packages --disable-sparsehash
-    - CCACHE_BASEDIR=$PWD make $MAKEOPTS
-    - make install
-    - export PYTHONPATH=$PWD/install/site-packages
-    - cd doc; python3 /usr/bin/sphinx-build -b doctest . build *.rst
-  tags:
-    - amd64
-
-job_clang_amd64_nosh:
-  script:
-    - ./autogen.sh
-    - ./configure CXX="ccache clang++" PYTHON=python3 --prefix=$PWD/install --with-python-module-path=$PWD/install/site-packages --disable-sparsehash
-    - CCACHE_BASEDIR=$PWD make $MAKEOPTS
-    - make install
-    - export PYTHONPATH=$PWD/install/site-packages
-    - cd doc; python3 /usr/bin/sphinx-build -b doctest . build *.rst
-  tags:
-    - amd64

doc/demos/cppextensions/Makefile

 CXX=g++
-CXXFLAGS=-O3 -fopenmp -std=gnu++17 -Wall -fPIC `pkg-config --cflags --libs graph-tool-py3.7` -shared
+CXXFLAGS=-O3 -fopenmp -std=gnu++17 -Wall -fPIC `pkg-config --cflags --libs graph-tool-py3.8` -shared
 ALL: libkcore.so
 
 libkcore.so: kcore.hh kcore.cc

doc/demos/cppextensions/cppextensions.rst

@@ -162,7 +162,7 @@ In graph-tool iteration can also be done in a more convenient way using
    }
 
 Both approaches above are equivalent. Graph-tool also provides
-``edges_range()``, ``out_edges_range()``, ``in_edges_range()``, etc. in
+``edges_range()``, ``out_edges_range()``, ``in_edges_range()``, etc., in
 an analogous fashion.
    
 Extracting specific property maps

doc/demos/inference/_edge_weights.rst

@@ -119,11 +119,15 @@ the weights, as follows:
 
    state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6),
               eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1),
-              edge_gradient=[], output="moreno-train-wsbm.svg")
+              edge_gradient=[], output="moreno-train-wsbm.pdf")
 
-.. figure:: moreno-train-wsbm.*
+.. testcleanup:: weighted-model
+
+   conv_png("moreno-train-wsbm.pdf")
+
+.. figure:: moreno-train-wsbm.png
    :align: center
-   :width: 350px
+   :width: 450px
 
    Best fit of the Binomial-weighted degree-corrected SBM for a network
    of terror suspects, using the strength of connection as edge
@@ -154,7 +158,7 @@ follows:
        os.chdir("demos/inference")
    except FileNotFoundError:
        pass
-   gt.seed_rng(44)
+   gt.seed_rng(44 + 4)
          
 .. testcode:: food-web
 
@@ -169,11 +173,15 @@ follows:
 
    state.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),
               eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True),
-              edge_gradient=[], output="foodweb-wsbm.svg")
+              edge_gradient=[], output="foodweb-wsbm.pdf")
+
+.. testcleanup:: food-web
 
-.. figure:: foodweb-wsbm.*
+   conv_png("foodweb-wsbm.pdf")
+              
+.. figure:: foodweb-wsbm.png
    :align: center
-   :width: 350px
+   :width: 450px
 
    Best fit of the exponential-weighted degree-corrected SBM for a food
    web, using the energy flow as edge covariates (indicated by the edge
@@ -204,11 +212,15 @@ can fit this alternative model simply by using the transformed weights:
 
    state_ln.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),
                  eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True),
-                 edge_gradient=[], output="foodweb-wsbm-lognormal.svg")
+                 edge_gradient=[], output="foodweb-wsbm-lognormal.pdf")
+
+.. testcleanup:: food-web
 
-.. figure:: foodweb-wsbm-lognormal.*
+   conv_png("foodweb-wsbm-lognormal.pdf")
+                 
+.. figure:: foodweb-wsbm-lognormal.png
    :align: center
-   :width: 350px
+   :width: 450px
 
    Best fit of the log-normal-weighted degree-corrected SBM for a food
    web, using the energy flow as edge covariates (indicated by the edge
@@ -245,9 +257,9 @@ Therefore, we can compute the posterior odds ratio between both models as:
 .. testoutput:: food-web
    :options: +NORMALIZE_WHITESPACE
 
-   ln Λ:  -24.246389...
+   ln Λ:  -16.814693...
 
-A value of :math:`\Lambda \approx \mathrm{e}^{-24} \approx 10^{-10}` in
+A value of :math:`\Lambda \approx \mathrm{e}^{-17} \approx 10^{-7}` in
 favor the exponential model indicates that the log-normal model does not
 provide a better fit for this particular data. Based on this, we
 conclude that the exponential model should be preferred in this case.

doc/demos/inference/_minimization.rst

@@ -21,7 +21,7 @@ network of American football teams, which we load from the
       os.chdir("demos/inference")
    except FileNotFoundError:
        pass
-   gt.seed_rng(8)
+   gt.seed_rng(12)
 
 .. testcode:: football
 
@@ -32,7 +32,7 @@ which yields
 
 .. testoutput:: football
 
-   <Graph object, undirected, with 115 vertices and 613 edges at 0x...>
+   <Graph object, undirected, with 115 vertices and 613 edges, 4 internal vertex properties, 2 internal graph properties, at 0x...>
 
 we then fit the degree-corrected model by calling
 
@@ -134,7 +134,7 @@ illustrate its use with the neural network of the `C. elegans
 
 .. testsetup:: celegans
 
-   gt.seed_rng(51)
+   gt.seed_rng(52)
 
 .. testcode:: celegans
 
@@ -145,7 +145,7 @@ which has 297 vertices and 2359 edges.
 
 .. testoutput:: celegans
 
-   <Graph object, directed, with 297 vertices and 2359 edges at 0x...>
+   <Graph object, directed, with 297 vertices and 2359 edges, 2 internal vertex properties, 1 internal edge property, 2 internal graph properties, at 0x...>
 
 A hierarchical fit of the degree-corrected model is performed as follows.
 
@@ -161,10 +161,16 @@ clustering using the
 
 .. testcode:: celegans
 
-   state.draw(output="celegans-hsbm-fit.svg")
+   state.draw(output="celegans-hsbm-fit.pdf")
 
-.. figure:: celegans-hsbm-fit.*
+.. testcleanup:: celegans
+
+   conv_png("celegans-hsbm-fit.pdf")
+                 
+
+.. figure:: celegans-hsbm-fit.png
    :align: center
+   :width: 80%
 
    Most likely hierarchical partition of the neural network of
    the *C. elegans* worm according to the nested degree-corrected SBM.
@@ -186,10 +192,10 @@ which shows the number of nodes and groups in all levels:
 
 .. testoutput:: celegans
 
-   l: 0, N: 297, B: 16
-   l: 1, N: 16, B: 8
-   l: 2, N: 8, B: 3
-   l: 3, N: 3, B: 1
+   l: 0, N: 297, B: 19
+   l: 1, N: 19, B: 6
+   l: 2, N: 6, B: 2
+   l: 3, N: 2, B: 1
 
 The hierarchical levels themselves are represented by individual
 :meth:`~graph_tool.inference.blockmodel.BlockState` instances obtained via the
@@ -203,10 +209,10 @@ The hierarchical levels themselves are represented by individual
 
 .. testoutput:: celegans
 
-    <BlockState object with 16 blocks (16 nonempty), degree-corrected, for graph <Graph object, directed, with 297 vertices and 2359 edges at 0x...>, at 0x...>
-    <BlockState object with 8 blocks (8 nonempty), for graph <Graph object, directed, with 16 vertices and 134 edges at 0x...>, at 0x...>
-    <BlockState object with 3 blocks (3 nonempty), for graph <Graph object, directed, with 8 vertices and 50 edges at 0x...>, at 0x...>
-    <BlockState object with 1 blocks (1 nonempty), for graph <Graph object, directed, with 3 vertices and 8 edges at 0x...>, at 0x...>
+   <BlockState object with 19 blocks (19 nonempty), degree-corrected, for graph <Graph object, directed, with 297 vertices and 2359 edges, 2 internal vertex properties, 1 internal edge property, 2 internal graph properties, at 0x...>, at 0x...>
+   <BlockState object with 6 blocks (6 nonempty), for graph <Graph object, directed, with 19 vertices and 176 edges, 2 internal vertex properties, 1 internal edge property, at 0x...>, at 0x...>
+   <BlockState object with 2 blocks (2 nonempty), for graph <Graph object, directed, with 6 vertices and 30 edges, 2 internal vertex properties, 1 internal edge property, at 0x...>, at 0x...>
+   <BlockState object with 1 blocks (1 nonempty), for graph <Graph object, directed, with 2 vertices and 4 edges, 2 internal vertex properties, 1 internal edge property, at 0x...>, at 0x...>
 
 This means that we can inspect the hierarchical partition just as before:
 
@@ -221,6 +227,54 @@ This means that we can inspect the hierarchical partition just as before:
 
 .. testoutput:: celegans
 
+   5
    2
-   1
    0
+
+Trade-off between memory usage and computation time
++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+The agglomerative algorithm behind
+:func:`~graph_tool.inference.minimize.minimize_blockmodel_dl` and
+:func:`~graph_tool.inference.minimize.minimize_nested_blockmodel_dl` has
+a log-linear complexity on the size of the network, but it makes several
+copies of the internal blockmodel state, which can become a problem for
+very large networks (i.e. tens of millions of edges or more). An
+alternative is to use a greedy algorithm based on a merge-split MCMC
+with zero temperature [peixoto-merge-split-2020]_, which requires a
+single global state, and thus can reduce memory usage. This is achieved
+by following the instructions in Sec. :ref:`sampling`, while setting the
+inverse temperature parameter ``beta`` to infinity. For example, an
+equivalent to the above minimization for the `C. elegans` network is the
+following:
+
+.. testcode:: celegans-mcmc
+
+   g = gt.collection.data["celegansneural"]
+
+   state = gt.NestedBlockState(g)
+
+   for i in range(1000): # this should be sufficiently large
+       state.multiflip_mcmc_sweep(beta=np.inf, niter=10)
+
+Whenever possible, this procedure should be repeated several times, and
+the result with the smallest description length (obtained via the
+:meth:`~graph_tool.inference.blockmodel.BlockState.entropy` method)
+should be chosen. Better results still can be obtained, at the expense
+of a longer computation time, by using the
+:meth:`~graph_tool.inference.mcmc.mcmc_anneal` function, which
+implements `simulated annealing
+<https://en.wikipedia.org/wiki/Simulated_annealing>`_:
+
+.. testcode:: celegans-mcmc-anneal
+
+   g = gt.collection.data["celegansneural"]
+
+   state = gt.NestedBlockState(g)
+
+   gt.mcmc_anneal(state, beta_range=(1, 10), niter=1000, mcmc_equilibrate_args=dict(force_niter=10))
+
+Any of the above methods should give similar results to the previous
+algorithms, while requiring less memory. In terms of quality of the
+results, it will vary depending on the data, thus experimentation is
+recommended.
\ No newline at end of file

doc/demos/inference/_model_class_selection.rst

@@ -154,8 +154,8 @@ evidence efficiently, as we show below, using
 
 .. testoutput:: model-evidence
 
-   Model evidence for deg_corr = True: -579.300446... (mean field), -832.245049... (Bethe)
-   Model evidence for deg_corr = False: -597.211055... (mean field), -728.704043... (Bethe)
+   Model evidence for deg_corr = True: -383.7870425700103 (mean field), -1053.691486781025 (Bethe)
+   Model evidence for deg_corr = False: -364.121582158287 (mean field), -945.9676904776013 (Bethe)
 
 If we consider the more accurate approximation, the outcome shows a
 preference for the non-degree-corrected model.
@@ -219,8 +219,8 @@ approach for the same network, using the nested model.
 
 .. testoutput:: model-evidence
 
-   Model evidence for deg_corr = True: -503.011203... (mean field), -728.064049... (Bethe)
-   Model evidence for deg_corr = False: -518.022407... (mean field), -633.507167... (Bethe)
+   Model evidence for deg_corr = True: -189.190149... (mean field), -844.518064... (Bethe)
+   Model evidence for deg_corr = False: -177.192565... (mean field), -727.585051... (Bethe)
 
 The results are similar: If we consider the most accurate approximation,
 the non-degree-corrected model possesses the largest evidence. Note also

doc/demos/inference/_model_selection.rst

@@ -25,8 +25,8 @@ we have
 .. testoutput:: model-selection
    :options: +NORMALIZE_WHITESPACE
 
-   Non-degree-corrected DL:	 8470.198169...
-   Degree-corrected DL:	 8273.840277...
+   Non-degree-corrected DL:	 8553.474528...
+   Degree-corrected DL:	 8266.554118...
    
 Since it yields the smallest description length, the degree-corrected
 fit should be preferred. The statistical significance of the choice can
@@ -52,13 +52,13 @@ fits. In our particular case, we have
 .. testoutput:: model-selection
    :options: +NORMALIZE_WHITESPACE
 
-   ln Λ:  -196.357892...
+   ln Λ:  -286.92041...
 
 The precise threshold that should be used to decide when to `reject a
 hypothesis <https://en.wikipedia.org/wiki/Hypothesis_testing>`_ is
 subjective and context-dependent, but the value above implies that the
-particular degree-corrected fit is around :math:`\mathrm{e}^{196}
-\approx 10^{85}` times more likely than the non-degree corrected one,
+particular degree-corrected fit is around :math:`\mathrm{e}^{287}
+\approx 10^{124}` times more likely than the non-degree corrected one,
 and hence it can be safely concluded that it provides a substantially
 better fit.
 
@@ -80,11 +80,11 @@ example, for the American football network above, we have:
 .. testoutput:: model-selection
    :options: +NORMALIZE_WHITESPACE
 
-   Non-degree-corrected DL:     1733.525685...
+   Non-degree-corrected DL:     1738.138494...
    Degree-corrected DL:         1780.576716...
-   ln Λ:                         -47.051031...
+   ln Λ:                        -42.4382219...
 
-Hence, with a posterior odds ratio of :math:`\Lambda \approx \mathrm{e}^{-47} \approx
-10^{-20}` in favor of the non-degree-corrected model, we conclude that the
+Hence, with a posterior odds ratio of :math:`\Lambda \approx \mathrm{e}^{-42} \approx
+10^{-18}` in favor of the non-degree-corrected model, we conclude that the
 degree-corrected variant is an unnecessarily complex description for
 this network.

doc/demos/inference/_prediction.rst

@@ -142,8 +142,8 @@ above).
 
 .. testoutput:: missing-edges
 
-   likelihood-ratio for (101, 102): 0.37...
-   likelihood-ratio for (17, 56): 0.62...
+   likelihood-ratio for (101, 102): 0.36...
+   likelihood-ratio for (17, 56): 0.63...
 
 From which we can conclude that edge :math:`(17, 56)` is more likely
 than :math:`(101, 102)` to be a missing edge.

doc/demos/inference/_reconstruction.rst

@@ -132,8 +132,8 @@ simple example, using
       os.chdir("demos/inference")
    except FileNotFoundError:
        pass
-   np.random.seed(42)
-   gt.seed_rng(44)
+   np.random.seed(43)
+   gt.seed_rng(45)
 
 .. testcode:: measured
 
@@ -151,14 +151,11 @@ simple example, using
    n[e] = 2                 # pretend we have measured non-edge (15, 73) twice,
    x[e] = 1                 # but observed it as an edge once.
 
-   bs = [g.get_vertices()] + [zeros(1)] * 5  # initial hierarchical partition
-
    # We inititialize MeasuredBlockState, assuming that each non-edge has
    # been measured only once (as opposed to twice for the observed
    # edges), as specified by the 'n_default' and 'x_default' parameters.
 
-   state = gt.MeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0,
-                                 state_args=dict(bs=bs))
+   state = gt.MeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0)
 
    # We will first equilibrate the Markov chain
    gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))
@@ -191,9 +188,9 @@ Which yields the following output:
    
 .. testoutput:: measured
 
-   Posterior probability of edge (11, 36): 0.742674...
-   Posterior probability of non-edge (15, 73): 0.020702...
-   Estimated average local clustering: 0.572384 ± 0.003466...
+   Posterior probability of edge (11, 36): 0.768976...
+   Posterior probability of non-edge (15, 73): 0.039203...
+   Estimated average local clustering: 0.571939 ± 0.003534...
 
 We have a successful reconstruction, where both ambiguous adjacency
 matrix entries are correctly recovered. The value for the average
@@ -287,8 +284,7 @@ with uniform error rates, as we see with the same example:
 
 .. testcode:: measured
 
-   state = gt.MixedMeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0,
-                                      state_args=dict(bs=bs))
+   state = gt.MixedMeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0)
 
    # We will first equilibrate the Markov chain
    gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))
@@ -312,9 +308,9 @@ Which yields:
    
 .. testoutput:: measured
 
-   Posterior probability of edge (11, 36): 0.839483...
-   Posterior probability of non-edge (15, 73): 0.061106...
-   Estimated average local clustering: 0.572476 ± 0.004370...
+   Posterior probability of edge (11, 36): 0.631563...
+   Posterior probability of non-edge (15, 73): 0.022402...
+   Estimated average local clustering: 0.570065 ± 0.007145...
 
 The results are very similar to the ones obtained with the uniform model
 in this case, but can be quite different in situations where a large
@@ -401,15 +397,13 @@ inference:
    e = g.add_edge(15, 73)
    q[e] = .5                     # ambiguous spurious edge
    
-   bs = [g.get_vertices()] + [zeros(1)] * 5  # initial hierarchical partition
-
    # We inititialize UncertainBlockState, assuming that each non-edge
    # has an uncertainty of q_default, chosen to preserve the expected
    # density of the original network:
 
    q_default = (E - q.a.sum()) / ((N * (N - 1))/2 - E)
    
-   state = gt.UncertainBlockState(g, q=q, q_default=q_default, state_args=dict(bs=bs))
+   state = gt.UncertainBlockState(g, q=q, q_default=q_default)
 
    # We will first equilibrate the Markov chain
    gt.mcmc_equilibrate(state, wait=2000, mcmc_args=dict(niter=10))
@@ -441,9 +435,9 @@ The above yields the output:
    
 .. testoutput:: uncertain
 
-   Posterior probability of edge (11, 36): 0.891889...
-   Posterior probability of non-edge (15, 73): 0.026002...
-   Estimated average local clustering: 0.544468 ± 0.021562...
+   Posterior probability of edge (11, 36): 0.775777...
+   Posterior probability of non-edge (15, 73): 0.010001...
+   Estimated average local clustering: 0.523013 ± 0.017268...
 
 The reconstruction is accurate, despite the two ambiguous entries having
 the same measurement probability. The reconstructed network is visualized below.
@@ -488,4 +482,74 @@ the same measurement probability. The reconstructed network is visualized below.
    reconstruction. The pie fractions on the nodes correspond to the
    probability of being in group associated with the respective color.
 
+
+Latent Poisson multigraphs
+++++++++++++++++++++++++++
+
+Even in situations where measurement errors can be neglected, it can
+still be useful to assume a given network is the outcome of a "hidden"
+multigraph model, i.e. more than one edge between nodes is allowed, but
+then its multiedges are "erased" by transforming them into simple
+edges. In this way, it is possible to construct generative models that
+can better handle situations where the underlying network possesses
+heterogeneous density, such as strong community structure and broad
+degree distributions [peixoto-latent-2020]_. This can be incorporated
+into the scheme of Eq. :eq:`posterior-reconstruction` by considering
+the data to be the observed simple graph,
+:math:`\boldsymbol{\mathcal{D}} = \boldsymbol G`.  We proceed in same
+way as in the previous reconstruction scenarios, but using instead
+:class:`~graph_tool.inference.uncertain_blockmodel.LatentMultigraphBlockState`.
+
+For example, in the following we will obtain the community structure and
+latent multiedges of a network of political books:
+
+.. testcode:: erased-poisson
+
+   g = gt.collection.data["polbooks"]
+
+   state = gt.LatentMultigraphBlockState(g)
+
+   # We will first equilibrate the Markov chain
+   gt.mcmc_equilibrate(state, wait=100, mcmc_args=dict(niter=10))
+
+   # Now we collect the marginals for exactly 100,000 sweeps, at
+   # intervals of 10 sweeps:
+
+   u = None              # marginal posterior multigraph
+   pv = None             # marginal posterior group membership probabilities
+   
+   def collect_marginals(s):
+      global pv, u
+      u = s.collect_marginal_multigraph(u)
+      bstate = state.get_block_state()
+      b = gt.perfect_prop_hash([bstate.levels[0].b])[0] 
+      pv = bstate.levels[0].collect_vertex_marginals(pv, b=b)
+
+   gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
+                       callback=collect_marginals)
+
+   # compute average multiplicities
+
+   ew = u.new_ep("double")
+   w = u.ep.w
+   wcount = u.ep.wcount
+   for e in u.edges():
+       ew[e] = (wcount[e].a * w[e].a).sum() / wcount[e].a.sum()
+   
+   bstate = state.get_block_state()
+   bstate = bstate.levels[0].copy(g=u)
+   pv = u.own_property(pv)
+   bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv,
+               edge_pen_width=gt.prop_to_size(ew, .1, 8, power=1), edge_gradient=None,
+               output="polbooks-erased-poisson.svg")
+   
+.. figure:: polbooks-erased-poisson.*
+   :align: center
+   :width: 450px
+
+   Reconstructed latent Poisson degree-corrected SBM for a network of
+   political books, showing the marginal mean edge multiplicities as
+   line thickness.  The pie fractions on the nodes correspond to the
+   probability of being in group associated with the respective color.
+
 .. include:: _reconstruction_dynamics.rst

doc/demos/inference/_reconstruction_dynamics.rst

@@ -111,7 +111,7 @@ epidemic process.
 
    # Create reconstruction state
    rstate = gt.EpidemicsBlockState(u, s=ss, beta=None, r=1e-6, global_beta=.1,  
-                                   state_args=dict(B=1), nested=False, aE=g.num_edges()) 
+                                   nested=False, aE=g.num_edges()) 
  
    # Now we collect the marginals for exactly 100,000 sweeps, at 
    # intervals of 10 sweeps: 
@@ -143,8 +143,8 @@ The reconstruction can accurately recover the hidden network and the infection p
                  
 .. testoutput:: dynamics
 
-   Posterior similarity:  0.978091...
-   Inferred infection probability: 0.68809 ± 0.054207
+   Posterior similarity:  0.990587...
+   Inferred infection probability: 0.692324 ± 0.0496223
 
 The figure below shows the reconstructed network and the inferred community structure.   
                  

doc/demos/inference/_sampling.rst

@@ -27,62 +27,44 @@ large networks.
 
 In order to perform such moves, one needs again to operate with
 :class:`~graph_tool.inference.blockmodel.BlockState` or
-:class:`~graph_tool.inference.nested_blockmodel.NestedBlockState` instances, and calling
-their :meth:`~graph_tool.inference.blockmodel.BlockState.mcmc_sweep` methods. For
-example, the following will perform 1000 sweeps of the algorithm with
-the network of characters in the novel Les Misérables, starting from a
-random partition into 20 groups
+:class:`~graph_tool.inference.nested_blockmodel.NestedBlockState`
+instances, and calling either their
+:meth:`~graph_tool.inference.blockmodel.BlockState.mcmc_sweep` or
+:meth:`~graph_tool.inference.blockmodel.BlockState.multiflip_mcmc_sweep`
+methods. The former implements a simpler MCMC where a single node is
+moved at a time, where the latter is a more efficient version that
+performs merges and splits [peixoto-merge-split-2020]_, which should be
+in general preferred.
+
+For example, the following will perform 1000 sweeps of the algorithm
+with the network of characters in the novel Les Misérables, starting
+from a random partition into 20 groups
 
 .. testcode:: model-averaging
 
    g = gt.collection.data["lesmis"]
 
-   state = gt.BlockState(g, B=20)   # This automatically initializes the state
-                                    # with a random partition into B=20
-                                    # nonempty groups; The user could
-                                    # also pass an arbitrary initial
-                                    # partition using the 'b' parameter.
+   state = gt.BlockState(g, B=1)   # This automatically initializes the state with a partition
+                                   # into one group. The user could also pass a higher number
+                                   # to start with a random partition of a given size, or pass a
+                                   # specific initial partition using the 'b' parameter.
 
    # Now we run 1,000 sweeps of the MCMC. Note that the number of groups
    # is allowed to change, so it will eventually move from the initial
-   # value of B=20 to whatever is most appropriate for the data.
+   # value of B=1 to whatever is most appropriate for the data.
 
-   dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)
+   dS, nattempts, nmoves = state.multiflip_mcmc_sweep(niter=1000)
 
    print("Change in description length:", dS)
    print("Number of accepted vertex moves:", nmoves)
 
 .. testoutput:: model-averaging
 
-   Change in description length: -353.848032...
-   Number of accepted vertex moves: 37490
+   Change in description length: -71.707346...
+   Number of accepted vertex moves: 44055
 
-.. note::
-
-   Starting from a random partition is rarely the best option, since it
-   may take a long time for it to equilibrate. It was done above simply
-   as an illustration on how to initialize
-   :class:`~graph_tool.inference.blockmodel.BlockState` by hand. Instead, a much
-   better option in practice is to start from an approximation to the
-   "ground state" obtained with
-   :func:`~graph_tool.inference.minimize.minimize_blockmodel_dl`, e.g.
-
-    .. testcode:: model-averaging
-
-       state = gt.minimize_blockmodel_dl(g)
-       state = state.copy(B=g.num_vertices())
-       dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)
-
-       print("Change in description length:", dS)
-       print("Number of accepted vertex moves:", nmoves)
-
-    .. testoutput:: model-averaging
-
-       Change in description length: 31.622518...
-       Number of accepted vertex moves: 43152
-
-Although the above is sufficient to implement model averaging, there is a
-convenience function called
+Although the above is sufficient to implement sampling from the
+posterior, there is a convenience function called
 :func:`~graph_tool.inference.mcmc.mcmc_equilibrate` that is intend to
 simplify the detection of equilibration, by keeping track of the maximum
 and minimum values of description length encountered and how many sweeps
@@ -93,58 +75,7 @@ have been made without a "record breaking" event. For example,
    # We will accept equilibration if 10 sweeps are completed without a
    # record breaking event, 2 consecutive times.
 
-   gt.mcmc_equilibrate(state, wait=10, nbreaks=2, mcmc_args=dict(niter=10), verbose=True)
-
-will output:
-
-.. testoutput:: model-averaging
-    :options: +NORMALIZE_WHITESPACE
-