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.. _inference-howto:

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Inferring modular network structure
===================================
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``graph-tool`` includes algorithms to identify the large-scale structure
of networks in the :mod:`~graph_tool.inference` submodule. Here we
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explain the basic functionality with self-contained examples. For a more
thorough theoretical introduction to the methods described here, the
reader is referred to [peixoto-bayesian-2017]_.
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Background: Nonparametric statistical inference
-----------------------------------------------

A common task when analyzing networks is to characterize their
structures in simple terms, often by dividing the nodes into modules or
"communities".

A principled approach to perform this task is to formulate `generative
models <https://en.wikipedia.org/wiki/Generative_model>`_ that include
the idea of "modules" in their descriptions, which then can be detected
by `inferring <https://en.wikipedia.org/wiki/Statistical_inference>`_
the model parameters from data. More precisely, given the partition
:math:`\boldsymbol b = \{b_i\}` of the network into :math:`B` groups,
where :math:`b_i\in[0,B-1]` is the group membership of node :math:`i`,
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we define a model that generates a network :math:`\boldsymbol G` with a
probability
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.. math::
   :label: model-likelihood

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   P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)
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where :math:`\boldsymbol\theta` are additional model parameters that
control how the node partition affects the structure of the
network. Therefore, if we observe a network :math:`\boldsymbol G`, the
likelihood that it was generated by a given partition :math:`\boldsymbol
b` is obtained via the `Bayesian
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<https://en.wikipedia.org/wiki/Bayesian_inference>`_ posterior

.. math::
   :label: model-posterior-sum

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   P(\boldsymbol b | \boldsymbol G) = \frac{\sum_{\boldsymbol\theta}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}
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where :math:`P(\boldsymbol\theta, \boldsymbol b)` is the `prior
probability <https://en.wikipedia.org/wiki/Prior_probability>`_ of the
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model parameters, and

.. math::
   :label: model-evidence

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   P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)
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is called the `evidence`. The particular types of model that will be
considered here have "hard constraints", such that there is only one
choice for the remaining parameters :math:`\boldsymbol\theta` that is
compatible with the generated network, such that
Eq. :eq:`model-posterior-sum` simplifies to
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.. math::
   :label: model-posterior

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   P(\boldsymbol b | \boldsymbol G) = \frac{P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}
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with :math:`\boldsymbol\theta` above being the only choice compatible with
:math:`\boldsymbol G` and :math:`\boldsymbol b`. The inference procedures considered
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here will consist in either finding a network partition that maximizes
Eq. :eq:`model-posterior`, or sampling different partitions according
its posterior probability.

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As we will show below, this approach also enables the comparison of
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`different` models according to statistical evidence (a.k.a. `model
selection`).

Minimum description length (MDL)
++++++++++++++++++++++++++++++++

We note that Eq. :eq:`model-posterior` can be written as

.. math::

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   P(\boldsymbol b | \boldsymbol G) = \frac{\exp(-\Sigma)}{P(\boldsymbol G)}
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where

.. math::
   :label: model-dl

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   \Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b)
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is called the **description length** of the network :math:`\boldsymbol
G`. It measures the amount of `information
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<https://en.wikipedia.org/wiki/Information_theory>`_ required to
describe the data, if we `encode
<https://en.wikipedia.org/wiki/Entropy_encoding>`_ it using the
particular parametrization of the generative model given by
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:math:`\boldsymbol\theta` and :math:`\boldsymbol b`, as well as the
parameters themselves. Therefore, if we choose to maximize the posterior
distribution of Eq. :eq:`model-posterior` it will be fully equivalent to
the so-called `minimum description length
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<https://en.wikipedia.org/wiki/Minimum_description_length>`_
method. This approach corresponds to an implementation of `Occam's razor
<https://en.wikipedia.org/wiki/Occam%27s_razor>`_, where the `simplest`
model is selected, among all possibilities with the same explanatory
power. The selection is based on the statistical evidence available, and
therefore will not `overfit
<https://en.wikipedia.org/wiki/Overfitting>`_, i.e. mistake stochastic
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fluctuations for actual structure. In particular this means that we will
not find modules in networks if they could have arisen simply because of
stochastic fluctuations, as they do in fully random graphs
[guimera-modularity-2004]_.
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The stochastic block model (SBM)
--------------------------------

The `stochastic block model
<https://en.wikipedia.org/wiki/Stochastic_block_model>`_ is arguably
the simplest generative process based on the notion of groups of
nodes [holland-stochastic-1983]_. The `microcanonical
<https://en.wikipedia.org/wiki/Microcanonical_ensemble>`_ formulation
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[peixoto-nonparametric-2017]_ of the basic or "traditional" version takes
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as parameters the partition of the nodes into groups
:math:`\boldsymbol b` and a :math:`B\times B` matrix of edge counts
:math:`\boldsymbol e`, where :math:`e_{rs}` is the number of edges
between groups :math:`r` and :math:`s`. Given these constraints, the
edges are then placed randomly. Hence, nodes that belong to the same
group possess the same probability of being connected with other
nodes of the network.

An example of a possible parametrization is given in the following
figure.

.. testcode:: sbm-example
   :hide:

   import os
   try:
      os.chdir("demos/inference")
   except FileNotFoundError:
       pass

   g = gt.load_graph("blockmodel-example.gt.gz")
   gt.graph_draw(g, pos=g.vp.pos, vertex_size=10, vertex_fill_color=g.vp.bo,
                 vertex_color="#333333",
                 edge_gradient=g.new_ep("vector<double>", val=[0]),
                 output="sbm-example.svg")

   ers = g.gp.w

   from pylab import *
   figure()
   matshow(log(ers))
   xlabel("Group $r$")
   ylabel("Group $s$")
   gca().xaxis.set_label_position("top") 
   savefig("sbm-example-ers.svg")

.. table::
    :class: figure

    +----------------------------------+------------------------------+
    |.. figure:: sbm-example-ers.svg   |.. figure:: sbm-example.svg   |
    |   :width: 300px                  |   :width: 300px              |
    |   :align: center                 |   :align: center             |
    |                                  |                              |
    |   Matrix of edge counts          |   Generated network.         |
    |   :math:`\boldsymbol e` between  |                              |
    |   groups.                        |                              |
    +----------------------------------+------------------------------+

.. note::

   We emphasize that no constraints are imposed on what `kind` of
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   modular structure is allowed, as the matrix of edge counts :math:`e`
   is unconstrained. Hence, we can detect the putatively typical pattern
   of `"community structure"
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   <https://en.wikipedia.org/wiki/Community_structure>`_, i.e. when
   nodes are connected mostly to other nodes of the same group, if it
   happens to be the most likely network description, but we can also
   detect a large multiplicity of other patterns, such as `bipartiteness
   <https://en.wikipedia.org/wiki/Bipartite_graph>`_, core-periphery,
   and many others, all under the same inference framework.


Although quite general, the traditional model assumes that the edges are
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placed randomly inside each group, and because of this the nodes that
belong to the same group tend to have very similar degrees. As it turns
out, this is often a poor model for many networks, which possess highly
heterogeneous degree distributions. A better model for such networks is
called the `degree-corrected` stochastic block model
[karrer-stochastic-2011]_, and it is defined just like the traditional
model, with the addition of the degree sequence :math:`\boldsymbol k =
\{k_i\}` of the graph as an additional set of parameters (assuming again
a microcanonical formulation [peixoto-nonparametric-2017]_).
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The nested stochastic block model
+++++++++++++++++++++++++++++++++

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The regular SBM has a drawback when applied to large networks. Namely,
it cannot be used to find relatively small groups, as the maximum number
of groups that can be found scales as
:math:`B_{\text{max}}=O(\sqrt{N})`, where :math:`N` is the number of
nodes in the network, if Bayesian inference is performed
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[peixoto-parsimonious-2013]_. In order to circumvent this, we need to
replace the noninformative priors used by a hierarchy of priors and
hyperpriors, which amounts to a `nested SBM`, where the groups
themselves are clustered into groups, and the matrix :math:`e` of edge
counts are generated from another SBM, and so on recursively
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[peixoto-hierarchical-2014]_, as illustrated below.
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.. figure:: nested-diagram.*
   :width: 400px
   :align: center

   Example of a nested SBM with three levels.

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With this model, the maximum number of groups that can be inferred
scales as :math:`B_{\text{max}}=O(N/\log(N))`. In addition to being able
to find small groups in large networks, this model also provides a
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multilevel hierarchical description of the network. With such a
description, we can uncover structural patterns at multiple scales,
representing different levels of coarse-graining.
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Inferring the best partition
----------------------------

The simplest and most efficient approach is to find the best
partition of the network by maximizing Eq. :eq:`model-posterior`
according to some version of the model. This is obtained via the
functions :func:`~graph_tool.inference.minimize_blockmodel_dl` or
:func:`~graph_tool.inference.minimize_nested_blockmodel_dl`, which
employs an agglomerative multilevel `Markov chain Monte Carlo (MCMC)
<https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo>`_ algorithm
[peixoto-efficient-2014]_.

We focus first on the non-nested model, and we illustrate its use with a
network of American football teams, which we load from the
:mod:`~graph_tool.collection` module:

.. testsetup:: football

   import os
   try:
      os.chdir("demos/inference")
   except FileNotFoundError:
       pass
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   gt.seed_rng(7)
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.. testcode:: football

   g = gt.collection.data["football"]
   print(g)

which yields

.. testoutput:: football

   <Graph object, undirected, with 115 vertices and 613 edges at 0x...>

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we then fit the degree-corrected model by calling
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.. testcode:: football

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   state = gt.minimize_blockmodel_dl(g)
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This returns a :class:`~graph_tool.inference.BlockState` object that
includes the inference results.

.. note::

   The inference algorithm used is stochastic by nature, and may return
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   a different answer each time it is run. This may be due to the fact
   that there are alternative partitions with similar probabilities, or
   that the optimum is difficult to find. Note that the inference
   problem here is, in general, `NP-Hard
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   <https://en.wikipedia.org/wiki/NP-hardness>`_, hence there is no
   efficient algorithm that is guaranteed to always find the best
   answer.

   Because of this, typically one would call the algorithm many times,
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   and select the partition with the largest posterior probability of
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   Eq. :eq:`model-posterior`, or equivalently, the minimum description
   length of Eq. :eq:`model-dl`. The description length of a fit can be
   obtained with the :meth:`~graph_tool.inference.BlockState.entropy`
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   method. See also Sec. :ref:`sec_model_selection` below.
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We may perform a drawing of the partition obtained via the
:mod:`~graph_tool.inference.BlockState.draw` method, that functions as a
convenience wrapper to the :func:`~graph_tool.draw.graph_draw` function

.. testcode:: football

   state.draw(pos=g.vp.pos, output="football-sbm-fit.svg")

which yields the following image.

.. figure:: football-sbm-fit.*
   :align: center
   :width: 400px

   Stochastic block model inference of a network of American college
   football teams. The colors correspond to inferred group membership of
   the nodes.

We can obtain the group memberships as a
:class:`~graph_tool.PropertyMap` on the vertices via the
:mod:`~graph_tool.inference.BlockState.get_blocks` method:

.. testcode:: football

   b = state.get_blocks()
   r = b[10]   # group membership of vertex 10
   print(r)

which yields:

.. testoutput:: football

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We may also access the matrix of edge counts between groups via
:mod:`~graph_tool.inference.BlockState.get_matrix`

.. testcode:: football

   e = state.get_matrix()

   matshow(e.todense())
   savefig("football-edge-counts.svg")

.. figure:: football-edge-counts.*
   :align: center

   Matrix of edge counts between groups.

We may obtain the same matrix of edge counts as a graph, which has
internal edge and vertex property maps with the edge and vertex counts,
respectively:

.. testcode:: football

   bg = state.get_bg()
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   ers = state.mrs    # edge counts
   nr = state.wr      # node counts
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.. _sec_model_selection:

Hierarchical partitions
+++++++++++++++++++++++

The inference of the nested family of SBMs is done in a similar manner,
but we must use instead the
:func:`~graph_tool.inference.minimize_nested_blockmodel_dl` function. We
illustrate its use with the neural network of the `C. elegans
<https://en.wikipedia.org/wiki/Caenorhabditis_elegans>`_ worm:

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.. testsetup:: celegans

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   gt.seed_rng(47)
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.. testcode:: celegans

   g = gt.collection.data["celegansneural"]
   print(g)

which has 297 vertices and 2359 edges.

.. testoutput:: celegans

   <Graph object, directed, with 297 vertices and 2359 edges at 0x...>

A hierarchical fit of the degree-corrected model is performed as follows.

.. testcode:: celegans

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   state = gt.minimize_nested_blockmodel_dl(g)
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The object returned is an instance of a
:class:`~graph_tool.inference.NestedBlockState` class, which
encapsulates the results. We can again draw the resulting hierarchical
clustering using the
:meth:`~graph_tool.inference.NestedBlockState.draw` method:

.. testcode:: celegans

   state.draw(output="celegans-hsbm-fit.svg")

.. figure:: celegans-hsbm-fit.*
   :align: center

   Most likely hierarchical partition of the neural network of
   the C. elegans worm according to the nested degree-corrected SBM.

.. note::

   If the ``output`` parameter to
   :meth:`~graph_tool.inference.NestedBlockState.draw` is omitted, an
   interactive visualization is performed, where the user can re-order
   the hierarchy nodes using the mouse and pressing the ``r`` key.

A summary of the inferred hierarchy can be obtained with the
:meth:`~graph_tool.inference.NestedBlockState.print_summary` method,
which shows the number of nodes and groups in all levels:

.. testcode:: celegans

   state.print_summary()

.. testoutput:: celegans

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   l: 0, N: 297, B: 17
   l: 1, N: 17, B: 9
   l: 2, N: 9, B: 3
   l: 3, N: 3, B: 1
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The hierarchical levels themselves are represented by individual
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:meth:`~graph_tool.inference.BlockState` instances obtained via the
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:meth:`~graph_tool.inference.NestedBlockState.get_levels()` method:

.. testcode:: celegans

   levels = state.get_levels()
   for s in levels:
       print(s)

.. testoutput:: celegans

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   <BlockState object with 17 blocks (17 nonempty), degree-corrected, for graph <Graph object, directed, with 297 vertices and 2359 edges at 0x...>, at 0x...>
   <BlockState object with 9 blocks (9 nonempty), for graph <Graph object, directed, with 17 vertices and 156 edges at 0x...>, at 0x...>
   <BlockState object with 3 blocks (3 nonempty), for graph <Graph object, directed, with 9 vertices and 57 edges at 0x...>, at 0x...>
   <BlockState object with 1 blocks (1 nonempty), for graph <Graph object, directed, with 3 vertices and 9 edges at 0x...>, at 0x...>
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This means that we can inspect the hierarchical partition just as before:

.. testcode:: celegans

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   r = levels[0].get_blocks()[46]    # group membership of node 46 in level 0
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   print(r)
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   r = levels[0].get_blocks()[r]     # group membership of node 46 in level 1
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   print(r)
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   r = levels[0].get_blocks()[r]     # group membership of node 46 in level 2
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   print(r)

.. testoutput:: celegans

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   7
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   0
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   0
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.. _model_selection:
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Model selection
+++++++++++++++

As mentioned above, one can select the best model according to the
choice that yields the smallest description length. For instance, in
case of the `C. elegans` network we have

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.. testsetup:: model-selection

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   gt.seed_rng(43)
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.. testcode:: model-selection

   g = gt.collection.data["celegansneural"]

   state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False)
   state_dc  = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)

   print("Non-degree-corrected DL:\t", state_ndc.entropy())
   print("Degree-corrected DL:\t", state_dc.entropy())

.. testoutput:: model-selection
   :options: +NORMALIZE_WHITESPACE

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   Non-degree-corrected DL:     8456.994339...
   Degree-corrected DL:         8233.850036...
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Since it yields the smallest description length, the degree-corrected
fit should be preferred. The statistical significance of the choice can
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be accessed by inspecting the posterior odds ratio
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[peixoto-nonparametric-2017]_
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.. math::

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   \Lambda &= \frac{P(\boldsymbol b, \mathcal{H}_\text{NDC} | \boldsymbol G)}{P(\boldsymbol b, \mathcal{H}_\text{DC} | \boldsymbol G)} \\
           &= \frac{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{NDC})}{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{DC})}\times\frac{P(\mathcal{H}_\text{NDC})}{P(\mathcal{H}_\text{DC})} \\
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           &= \exp(-\Delta\Sigma)

where :math:`\mathcal{H}_\text{NDC}` and :math:`\mathcal{H}_\text{DC}`
correspond to the non-degree-corrected and degree-corrected model
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hypotheses (assumed to be equally likely `a priori`), respectively, and
:math:`\Delta\Sigma` is the difference of the description length of both
fits. In our particular case, we have
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.. testcode:: model-selection

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   print(u"ln \u039b: ", state_dc.entropy() - state_ndc.entropy())
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.. testoutput:: model-selection
   :options: +NORMALIZE_WHITESPACE

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   ln Λ:  -223.144303...
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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
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particular degree-corrected fit is around :math:`\mathrm{e}^{233} \approx 10^{96}`
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times more likely than the non-degree corrected one, and hence it can be
safely concluded that it provides a substantially better fit.

Although it is often true that the degree-corrected model provides a
better fit for many empirical networks, there are also exceptions. For
example, for the American football network above, we have:

.. testcode:: model-selection

   g = gt.collection.data["football"]

   state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False)
   state_dc  = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)

   print("Non-degree-corrected DL:\t", state_ndc.entropy())
   print("Degree-corrected DL:\t", state_dc.entropy())
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   print(u"ln \u039b:\t\t\t", state_ndc.entropy() - state_dc.entropy())
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.. testoutput:: model-selection
   :options: +NORMALIZE_WHITESPACE

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   Non-degree-corrected DL:     1734.814739...
   Degree-corrected DL:         1780.576716...
   ln Λ:                         -45.761977...
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Hence, with a posterior odds ratio of :math:`\Lambda \approx \mathrm{e}^{-45} \approx
10^{-19}` in favor of the non-degree-corrected model, it seems like the
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degree-corrected variant is an unnecessarily complex description for
this network.

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.. _sampling:

Sampling from the posterior distribution
----------------------------------------
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When analyzing empirical networks, one should be open to the possibility
that there will be more than one fit of the SBM with similar posterior
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probabilities. In such situations, one should instead `sample`
partitions from the posterior distribution, instead of simply finding
its maximum. One can then compute quantities that are averaged over the
different model fits, weighted according to their posterior
probabilities.
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Full support for model averaging is implemented in ``graph-tool`` via an
efficient `Markov chain Monte Carlo (MCMC)
<https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo>`_ algorithm
[peixoto-efficient-2014]_. It works by attempting to move nodes into
different groups with specific probabilities, and `accepting or
rejecting
<https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm>`_
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such moves so that, after a sufficiently long time, the partitions will
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be observed with the desired posterior probability. The algorithm is
designed so that its run-time (i.e. each sweep of the MCMC) is linear on
the number of edges in the network, and independent on the number of
groups being used in the model, and hence is suitable for use on very
large networks.
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In order to perform such moves, one needs again to operate with
:class:`~graph_tool.inference.BlockState` or
:class:`~graph_tool.inference.NestedBlockState` instances, and calling
their :meth:`~graph_tool.inference.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

.. 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.

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   # 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.
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   dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)
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   print("Change in description length:", dS)
   print("Number of accepted vertex moves:", nmoves)

.. testoutput:: model-averaging

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   Change in description length: -365.317522...
   Number of accepted vertex moves: 38213
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.. note::

   Starting from a random partition is rarely the best option, since it
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   may take a long time for it to equilibrate. It was done above simply
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   as an illustration on how to initialize
   :class:`~graph_tool.inference.BlockState` by hand. Instead, a much
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   better option in practice is to start from an approximation to the
   "ground state" obtained with
   :func:`~graph_tool.inference.minimize_blockmodel_dl`, e.g.
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    .. testcode:: model-averaging

       state = gt.minimize_blockmodel_dl(g)
       state = state.copy(B=g.num_vertices())
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       dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)
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       print("Change in description length:", dS)
       print("Number of accepted vertex moves:", nmoves)

    .. testoutput:: model-averaging

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       Change in description length: 1.660677...
       Number of accepted vertex moves: 40461
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Although the above is sufficient to implement model averaging, there is a
convenience function called
:func:`~graph_tool.inference.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
have been made without a "record breaking" event. For example,

.. testcode:: model-averaging

   # 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

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    niter:     1  count:    0  breaks:  0  min_S: 706.26857  max_S: 708.14483  S: 708.14483  ΔS:      1.87626  moves:   418 
    niter:     2  count:    0  breaks:  0  min_S: 699.23453  max_S: 708.14483  S: 699.23453  ΔS:     -8.91030  moves:   409 
    niter:     3  count:    0  breaks:  0  min_S: 699.23453  max_S: 715.33531  S: 715.33531  ΔS:      16.1008  moves:   414 
    niter:     4  count:    0  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 723.13301  ΔS:      7.79770  moves:   391 
    niter:     5  count:    1  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 702.93354  ΔS:     -20.1995  moves:   411 
    niter:     6  count:    2  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 706.39029  ΔS:      3.45675  moves:   389 
    niter:     7  count:    3  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 706.80859  ΔS:     0.418293  moves:   404 
    niter:     8  count:    4  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 707.61960  ΔS:     0.811010  moves:   417 
    niter:     9  count:    5  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 706.46577  ΔS:     -1.15383  moves:   392 
    niter:    10  count:    6  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 714.34671  ΔS:      7.88094  moves:   410 
    niter:    11  count:    7  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 706.43194  ΔS:     -7.91477  moves:   383 
    niter:    12  count:    8  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 705.19434  ΔS:     -1.23760  moves:   405 
    niter:    13  count:    9  breaks:  0  min_S: 699.23453  max_S: 723.13301  S: 702.21395  ΔS:     -2.98039  moves:   423 
    niter:    14  count:    0  breaks:  1  min_S: 715.54878  max_S: 715.54878  S: 715.54878  ΔS:      13.3348  moves:   400 
    niter:    15  count:    0  breaks:  1  min_S: 715.54878  max_S: 716.65842  S: 716.65842  ΔS:      1.10964  moves:   413 
    niter:    16  count:    0  breaks:  1  min_S: 701.19994  max_S: 716.65842  S: 701.19994  ΔS:     -15.4585  moves:   382 
    niter:    17  count:    1  breaks:  1  min_S: 701.19994  max_S: 716.65842  S: 715.56997  ΔS:      14.3700  moves:   394 
    niter:    18  count:    0  breaks:  1  min_S: 701.19994  max_S: 719.25577  S: 719.25577  ΔS:      3.68580  moves:   404 
    niter:    19  count:    0  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 723.78811  ΔS:      4.53233  moves:   413 
    niter:    20  count:    1  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 709.77340  ΔS:     -14.0147  moves:   387 
    niter:    21  count:    2  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 714.14891  ΔS:      4.37551  moves:   419 
    niter:    22  count:    3  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 722.05875  ΔS:      7.90984  moves:   399 
    niter:    23  count:    4  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 714.32503  ΔS:     -7.73371  moves:   422 
    niter:    24  count:    5  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 708.53927  ΔS:     -5.78576  moves:   392 
    niter:    25  count:    6  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 714.05889  ΔS:      5.51962  moves:   404 
    niter:    26  count:    7  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 713.93196  ΔS:    -0.126937  moves:   414 
    niter:    27  count:    8  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 709.49863  ΔS:     -4.43333  moves:   410 
    niter:    28  count:    9  breaks:  1  min_S: 701.19994  max_S: 723.78811  S: 707.42167  ΔS:     -2.07696  moves:   397 
    niter:    29  count:    0  breaks:  1  min_S: 699.89982  max_S: 723.78811  S: 699.89982  ΔS:     -7.52185  moves:   388 
    niter:    30  count:    0  breaks:  1  min_S: 698.57305  max_S: 723.78811  S: 698.57305  ΔS:     -1.32677  moves:   391 
    niter:    31  count:    1  breaks:  1  min_S: 698.57305  max_S: 723.78811  S: 706.02629  ΔS:      7.45324  moves:   412 
    niter:    32  count:    2  breaks:  1  min_S: 698.57305  max_S: 723.78811  S: 701.97778  ΔS:     -4.04852  moves:   421 
    niter:    33  count:    3  breaks:  1  min_S: 698.57305  max_S: 723.78811  S: 707.50134  ΔS:      5.52356  moves:   410 
    niter:    34  count:    4  breaks:  1  min_S: 698.57305  max_S: 723.78811  S: 708.56686  ΔS:      1.06552  moves:   424 
    niter:    35  count:    0  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 724.07361  ΔS:      15.5067  moves:   399 
    niter:    36  count:    1  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 723.51969  ΔS:    -0.553915  moves:   384 
    niter:    37  count:    2  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 702.36708  ΔS:     -21.1526  moves:   406 
    niter:    38  count:    3  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 707.60129  ΔS:      5.23420  moves:   405 
    niter:    39  count:    4  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 709.67542  ΔS:      2.07413  moves:   400 
    niter:    40  count:    5  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 714.52753  ΔS:      4.85212  moves:   398 
    niter:    41  count:    6  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 707.86563  ΔS:     -6.66190  moves:   409 
    niter:    42  count:    7  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 718.80926  ΔS:      10.9436  moves:   400 
    niter:    43  count:    8  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 716.37312  ΔS:     -2.43615  moves:   378 
    niter:    44  count:    9  breaks:  1  min_S: 698.57305  max_S: 724.07361  S: 713.76944  ΔS:     -2.60368  moves:   399 
    niter:    45  count:   10  breaks:  2  min_S: 698.57305  max_S: 724.07361  S: 715.29009  ΔS:      1.52066  moves:   421
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Note that the value of ``wait`` above was made purposefully low so that
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the output would not be overly long. The most appropriate value requires
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experimentation, but a typically good value is ``wait=1000``.
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The function :func:`~graph_tool.inference.mcmc_equilibrate` accepts a
``callback`` argument that takes an optional function to be invoked
after each call to
:meth:`~graph_tool.inference.BlockState.mcmc_sweep`. This function
should accept a single parameter which will contain the actual
:class:`~graph_tool.inference.BlockState` instance. We will use this in
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the example below to collect the posterior vertex marginals (via
:class:`~graph_tool.inference.BlockState.collect_vertex_marginals`),
i.e. the posterior probability that a node belongs to a given group:
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.. testcode:: model-averaging

   # We will first equilibrate the Markov chain
   gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))

   pv = None 

   def collect_marginals(s):
      global pv
      pv = s.collect_vertex_marginals(pv)

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   # Now we collect the marginals for exactly 100,000 sweeps, at
   # intervals of 10 sweeps:
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   gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
                       callback=collect_marginals)

   # Now the node marginals are stored in property map pv. We can
   # visualize them as pie charts on the nodes:
   state.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv,
              edge_gradient=None, output="lesmis-sbm-marginals.svg")

.. figure:: lesmis-sbm-marginals.*
   :align: center
   :width: 450px

   Marginal probabilities of group memberships of the network of
   characters in the novel Les Misérables, according to the
   degree-corrected SBM. The `pie fractions
   <https://en.wikipedia.org/wiki/Pie_chart>`_ on the nodes correspond
   to the probability of being in group associated with the respective
   color.

We can also obtain a marginal probability on the number of groups
itself, as follows.

.. testcode:: model-averaging

   h = np.zeros(g.num_vertices() + 1)

   def collect_num_groups(s):
       B = s.get_nonempty_B()
       h[B] += 1

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   # Now we collect the marginals for exactly 100,000 sweeps, at
   # intervals of 10 sweeps:
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   gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
                       callback=collect_num_groups)

.. testcode:: model-averaging
   :hide:

   figure()
   Bs = np.arange(len(h))
   idx = h > 0
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   bar(Bs[idx], h[idx] / h.sum(), width=1, color="#ccb974")
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   gca().set_xticks([6,7,8,9])
   xlabel("$B$")
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   ylabel(r"$P(B|\boldsymbol G)$")
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   savefig("lesmis-B-posterior.svg")

.. figure:: lesmis-B-posterior.*
   :align: center

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   Marginal posterior probability of the number of nonempty groups for
   the network of characters in the novel Les Misérables, according to
   the degree-corrected SBM.
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Hierarchical partitions
+++++++++++++++++++++++

We can also perform model averaging using the nested SBM, which will
give us a distribution over hierarchies. The whole procedure is fairly
analogous, but now we make use of
:class:`~graph_tool.inference.NestedBlockState` instances.

.. note::

    When using :class:`~graph_tool.inference.NestedBlockState` instances
    to perform model averaging, they need to be constructed with the
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    option ``sampling=True``.
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Here we perform the sampling of hierarchical partitions using the same
network as above.

.. testcode:: nested-model-averaging

   g = gt.collection.data["lesmis"]

   state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov
                                               # chain from the "ground
                                               # state"

   # Before doing model averaging, the need to create a NestedBlockState
   # by passing sampling = True.

   # We also want to increase the maximum hierarchy depth to L = 10

   # We can do both of the above by copying.

   bs = state.get_bs()                     # Get hierarchical partition.
   bs += [np.zeros(1)] * (10 - len(bs))    # Augment it to L = 10 with
                                           # single-group levels.

   state = state.copy(bs=bs, sampling=True)

   # Now we run 1000 sweeps of the MCMC

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   dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)
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   print("Change in description length:", dS)
   print("Number of accepted vertex moves:", nmoves)

.. testoutput:: nested-model-averaging

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   Change in description length: 2.371018...
   Number of accepted vertex moves: 56087
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Similarly to the the non-nested case, we can use
:func:`~graph_tool.inference.mcmc_equilibrate` to do most of the boring
work, and we can now obtain vertex marginals on all hierarchical levels:


.. testcode:: nested-model-averaging

   # We will first equilibrate the Markov chain
   gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))

   pv = [None] * len(state.get_levels())

   def collect_marginals(s):
      global pv
      pv = [sl.collect_vertex_marginals(pv[l]) for l, sl in enumerate(s.get_levels())]

   # Now we collect the marginals for exactly 100,000 sweeps
   gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
                       callback=collect_marginals)

   # Now the node marginals for all levels are stored in property map
   # list pv. We can visualize the first level as pie charts on the nodes:
   state_0 = state.get_levels()[0]
   state_0.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv[0],
                edge_gradient=None, output="lesmis-nested-sbm-marginals.svg")

.. figure:: lesmis-nested-sbm-marginals.*
   :align: center
   :width: 450px

   Marginal probabilities of group memberships of the network of
   characters in the novel Les Misérables, according to the nested
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   degree-corrected SBM. The pie fractions on the nodes correspond to
   the probability of being in group associated with the respective
   color.
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We can also obtain a marginal probability of the number of groups
itself, as follows.

.. testcode:: nested-model-averaging

   h = [np.zeros(g.num_vertices() + 1) for s in state.get_levels()]

   def collect_num_groups(s):
       for l, sl in enumerate(s.get_levels()):
          B = sl.get_nonempty_B()
          h[l][B] += 1

   # Now we collect the marginal distribution for exactly 100,000 sweeps
   gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
                       callback=collect_num_groups)

.. testcode:: nested-model-averaging
   :hide:

   figure()
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   f, ax = plt.subplots(1, 5, figsize=(10, 3))
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   for i, h_ in enumerate(h[:5]):
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       Bs = np.arange(len(h_))
       idx = h_ > 0
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       ax[i].bar(Bs[idx], h_[idx] / h_.sum(), width=1, color="#ccb974")
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       ax[i].set_xticks(Bs[idx])
       ax[i].set_xlabel("$B_{%d}$" % i)
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       ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)
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       locator = MaxNLocator(prune='both', nbins=5)
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       ax[i].yaxis.set_major_locator(locator)
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   tight_layout()
   savefig("lesmis-nested-B-posterior.svg")

.. figure:: lesmis-nested-B-posterior.*
   :align: center

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   Marginal posterior probability of the number of nonempty groups
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   :math:`B_l` at each hierarchy level :math:`l` for the network of
   characters in the novel Les Misérables, according to the nested
   degree-corrected SBM.

Below we obtain some hierarchical partitions sampled from the posterior
distribution.

.. testcode:: nested-model-averaging

   for i in range(10):
       state.mcmc_sweep(niter=1000)
       state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False)

.. image:: lesmis-partition-sample-0.svg
   :width: 200px
.. image:: lesmis-partition-sample-1.svg
   :width: 200px
.. image:: lesmis-partition-sample-2.svg
   :width: 200px
.. image:: lesmis-partition-sample-3.svg
   :width: 200px
.. image:: lesmis-partition-sample-4.svg
   :width: 200px
.. image:: lesmis-partition-sample-5.svg
   :width: 200px
.. image:: lesmis-partition-sample-6.svg
   :width: 200px
.. image:: lesmis-partition-sample-7.svg
   :width: 200px
.. image:: lesmis-partition-sample-8.svg
   :width: 200px
.. image:: lesmis-partition-sample-9.svg
   :width: 200px

Model class selection
+++++++++++++++++++++

When averaging over partitions, we may be interested in evaluating which
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**model class** provides a better fit of the data, considering all
possible parameter choices. This is done by evaluating the model
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evidence summed over all possible partitions [peixoto-nonparametric-2017]_:
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.. math::

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   P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G,\boldsymbol\theta, \boldsymbol b) =  \sum_{\boldsymbol b}P(\boldsymbol G,\boldsymbol b).
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This quantity is analogous to a `partition function
<https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)>`_
in statistical physics, which we can write more conveniently as a
negative `free energy
<https://en.wikipedia.org/wiki/Thermodynamic_free_energy>`_ by taking
its logarithm

.. math::
   :label: free-energy

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   \ln P(\boldsymbol G) = \underbrace{\sum_{\boldsymbol b}q(\boldsymbol b)\ln P(\boldsymbol G,\boldsymbol b)}_{-\left<\Sigma\right>}\;
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              \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}}

where

.. math::

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   q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}
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is the posterior probability of partition :math:`\boldsymbol b`. The
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first term of Eq. :eq:`free-energy` (the "negative energy") is minus the
average of description length :math:`\left<\Sigma\right>`, weighted
according to the posterior distribution. The second term
:math:`\mathcal{S}` is the `entropy
<https://en.wikipedia.org/wiki/Entropy_(information_theory)>`_ of the
posterior distribution, and measures, in a sense, the "quality of fit"
of the model: If the posterior is very "peaked", i.e. dominated by a
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single partition with a very large probability, the entropy will tend to
zero. However, if there are many partitions with similar probabilities
--- meaning that there is no single partition that describes the network
uniquely well --- it will take a large value instead.
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Since the MCMC algorithm samples partitions from the distribution
:math:`q(\boldsymbol b)`, it can be used to compute
:math:`\left<\Sigma\right>` easily, simply by averaging the description
length values encountered by sampling from the posterior distribution
many times.

The computation of the posterior entropy :math:`\mathcal{S}`, however,
is significantly more difficult, since it involves measuring the precise
value of :math:`q(\boldsymbol b)`. A direct "brute force" computation of
:math:`\mathcal{S}` is implemented via
:meth:`~graph_tool.inference.BlockState.collect_partition_histogram` and
:func:`~graph_tool.inference.microstate_entropy`, however this is only
feasible for very small networks. For larger networks, we are forced to
perform approximations. The simplest is a "mean field" one, where we
assume the posterior factorizes as

.. math::

   q(\boldsymbol b) \approx \prod_i{q_i(b_i)}

where

.. math::

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   q_i(r) = P(b_i = r | \boldsymbol G)
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is the marginal group membership distribution of node :math:`i`. This
yields an entropy value given by

.. math::

   S \approx -\sum_i\sum_rq_i(r)\ln q_i(r).

This approximation should be seen as an upper bound, since any existing
correlation between the nodes (which are ignored here) will yield
smaller entropy values.

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A more accurate assumption is called the `Bethe approximation`
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[mezard-information-2009]_, and takes into account the correlation
between adjacent nodes in the network,

.. math::

   q(\boldsymbol b) \approx \prod_{i<j}q_{ij}(b_i,b_j)^{A_{ij}}\prod_iq_i(b_i)^{1-k_i}

where :math:`A_{ij}` is the `adjacency matrix
<https://en.wikipedia.org/wiki/Adjacency_matrix>`_, :math:`k_i` is the
degree of node :math:`i`, and

.. math::

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   q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)
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is the joint group membership distribution of nodes :math:`i` and
:math:`j` (a.k.a. the `edge marginals`). This yields an entropy value
given by

.. math::

   S \approx -\sum_{i<j}A_{ij}\sum_{rs}q_{ij}(r,s)\ln q_{ij}(r,s) - \sum_i(1-k_i)\sum_rq_i(r)\ln q_i(r).

Typically, this approximation yields smaller values than the mean field
one, and is generally considered to be superior. However, formally, it
depends on the graph being sufficiently locally "tree-like", and the
posterior being indeed strongly correlated with the adjacency matrix
itself --- two characteristics which do not hold in general. Although
the approximation often gives reasonable results even when these
conditions do not strictly hold, in some situations when they are
strongly violated this approach can yield meaningless values, such as a
negative entropy. Therefore, it is useful to compare both approaches
whenever possible.

With these approximations, it possible to estimate the full model
evidence efficiently, as we show below, using
:meth:`~graph_tool.inference.BlockState.collect_vertex_marginals`,
:meth:`~graph_tool.inference.BlockState.collect_edge_marginals`,
:meth:`~graph_tool.inference.mf_entropy` and
:meth:`~graph_tool.inference.bethe_entropy`.

.. testcode:: model-evidence

   g = gt.collection.data["lesmis"]

   for deg_corr in [True, False]:
       state = gt.minimize_blockmodel_dl(g, deg_corr=deg_corr)     # Initialize the Markov
                                                                   # chain from the "ground
                                                                   # state"
       state = state.copy(B=g.num_vertices())

       dls = []         # description length history
       vm = None        # vertex marginals
       em = None        # edge marginals

       def collect_marginals(s):
           global vm, em
           vm = s.collect_vertex_marginals(vm)
           em = s.collect_edge_marginals(em)
           dls.append(s.entropy())

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       # Now we collect the marginal distributions for exactly 200,000 sweeps
       gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10),
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                           callback=collect_marginals)

       S_mf = gt.mf_entropy(g, vm)
       S_bethe = gt.bethe_entropy(g, em)[0]
       L = -mean(dls)

       print("Model evidence for deg_corr = %s:" % deg_corr,
             L + S_mf, "(mean field),", L + S_bethe, "(Bethe)")

.. testoutput:: model-evidence

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   Model evidence for deg_corr = True: -569.590426... (mean field), -817.788531... (Bethe)
   Model evidence for deg_corr = False: -587.028530... (mean field), -736.990655... (Bethe)
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If we consider the more accurate approximation, the outcome shows a
preference for the non-degree-corrected model.
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When using the nested model, the approach is entirely analogous. The
only difference now is that we have a hierarchical partition
:math:`\{\boldsymbol b_l\}` in the equations above, instead of simply
:math:`\boldsymbol b`. In order to make the approach tractable, we
assume the factorization

.. math::

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   q(\{\boldsymbol b_l\}) \approx \prod_lq_l(\boldsymbol b_l)
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where :math:`q_l(\boldsymbol b_l)` is the marginal posterior for the
partition at level :math:`l`. For :math:`q_0(\boldsymbol b_0)` we may
use again either the mean-field or Bethe approximations, however for
:math:`l>0` only the mean-field approximation is applicable, since the
adjacency matrix of the higher layers is not constant. We show below the
approach for the same network, using the nested model.


.. testcode:: model-evidence

   g = gt.collection.data["lesmis"]

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   nL = 10
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   for deg_corr in [True, False]:
       state = gt.minimize_nested_blockmodel_dl(g, deg_corr=deg_corr)     # Initialize the Markov
                                                                          # chain from the "ground
                                                                          # state"
       bs = state.get_bs()                     # Get hierarchical partition.
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       bs += [np.zeros(1)] * (nL - len(bs))    # Augment it to L = 10 with
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                                               # single-group levels.

       state = state.copy(bs=bs, sampling=True)

       dls = []                               # description length history
       vm = [None] * len(state.get_levels())  # vertex marginals
       em = None                              # edge marginals

       def collect_marginals(s):
           global vm, em
           levels = s.get_levels()
           vm = [sl.collect_vertex_marginals(vm[l]) for l, sl in enumerate(levels)]
           em = levels[0].collect_edge_marginals(em)
           dls.append(s.entropy())

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       # Now we collect the marginal distributions for exactly 200,000 sweeps
       gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10),
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                           callback=collect_marginals)

       S_mf = [gt.mf_entropy(sl.g, vm[l]) for l, sl in enumerate(state.get_levels())]
       S_bethe = gt.bethe_entropy(g, em)[0]
       L = -mean(dls)

       print("Model evidence for deg_corr = %s:" % deg_corr,
             L + sum(S_mf), "(mean field),", L + S_bethe + sum(S_mf[1:]), "(Bethe)")


.. testoutput:: model-evidence

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   Model evidence for deg_corr = True: -551.228195... (mean field), -740.460493... (Bethe)
   Model evidence for deg_corr = False: -544.660366... (mean field), -649.135026... (Bethe)
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The results are similar: If we consider the most accurate approximation,
the non-degree-corrected model possesses the largest evidence. Note also
that we observe a better evidence for the nested models themselves, when
comparing to the evidences for the non-nested model --- which is not
quite surprising, since the non-nested model is a special case of the
nested one.
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.. _weights:
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Edge weights and covariates
---------------------------
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Very often networks cannot be completely represented by simple graphs,
but instead have arbitrary "weights" :math:`x_{ij}` on the edges. Edge
weights can be continuous or discrete numbers, and either strictly
positive or positive or negative, depending on context. The SBM can be
extended to cover these cases by treating edge weights as covariates
that are sampled from some distribution conditioned on the node
partition [aicher-learning-2015]_ [peixoto-weighted-2017]_, i.e.

.. math::
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   P(\boldsymbol x,\boldsymbol G|\boldsymbol b) =
   P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b),
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where :math:`P(\boldsymbol G|\boldsymbol b)` is the likelihood of the
unweighted SBM described previously, and :math:`P(\boldsymbol
x|\boldsymbol G,\boldsymbol b)` is the integrated likelihood of the edge
weights

.. math::

   P(\boldsymbol x|\boldsymbol G,\boldsymbol b) =
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where :math:`P({\boldsymbol x}_{rs}|\gamma)` is some model for the weights
:math:`{\boldsymbol x}_{rs}` between groups :math:`(r,s)`, conditioned on
some parameter :math:`\gamma`, sampled from its prior
:math:`P(\gamma)`. A hierarchical version of the model can also be
implemented by replacing this prior by a nested sequence of priors and
hyperpriors, as described in [peixoto-weighted-2017]_. The posterior
partition distribution is then simply
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.. math::

   P(\boldsymbol b | \boldsymbol G,\boldsymbol x) =
   \frac{P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b)
         P(\boldsymbol b)}{P(\boldsymbol G,\boldsymbol x)},

which can be sampled from, or maximized, just like with the unweighted
case, but will use the information on the weights to guide the partitions.

A variety of weight models is supported, reflecting different kinds of
edge covariates:

.. csv-table::
   :header: "Name", "Domain", "Bounds", "Shape"
   :widths: 10, 5, 5, 5
   :delim: |
   :align: center

   ``"real-exponential"``   | Real    :math:`(\mathbb{R})` | :math:`[0,\infty]`       | `Exponential <https://en.wikipedia.org/wiki/Exponential_distribution>`_
   ``"real-normal"``        | Real    :math:`(\mathbb{R})` | :math:`[-\infty,\infty]` | `Normal <https://en.wikipedia.org/wiki/Normal_distribution>`_
   ``"discrete-geometric"`` | Natural :math:`(\mathbb{N})` | :math:`[0,\infty]`       | `Geometric <https://en.wikipedia.org/wiki/Geometric_distribution>`_
   ``"discrete-binomial"``  | Natural :math:`(\mathbb{N})` | :math:`[0,M]`            | `Binomial <https://en.wikipedia.org/wiki/Binomial_distribution>`_
   ``"discrete-poisson"``   | Natural :math:`(\mathbb{N})` | :math:`[0,\infty]`       | `Poisson <https://en.wikipedia.org/wiki/Poisson_distribution>`_

In fact, the actual model implements `microcanonical
<https://en.wikipedia.org/wiki/Microcanonical_ensemble>`_ versions of
these distributions that are asymptotically equivalent, as described in
[peixoto-weighted-2017]_. These can be combined with arbitrary weight
transformations to achieve a large family of associated
distributions. For example, to use a `log-normal
<https://en.wikipedia.org/wiki/Log-normal_distribution>`_ weight model
for positive real weights :math:`\boldsymbol x`, we can use the
transformation :math:`y_{ij} = \ln x_{ij}` together with the
``"real-normal"`` model for :math:`\boldsymbol y`. To model weights that
are positive or negative integers in :math:`\mathbb{Z}`, we could either
subtract the minimum value, :math:`y_{ij} = x_{ij} - x^*`, with
:math:`x^*=\operatorname{min}_{ij}x_{ij}`, and use any of the above
models for non-negative integers in :math:`\mathbb{N}`, or
alternatively, consider the sign as an additional covariate,
i.e. :math:`s_{ij} = [\operatorname{sign}(x_{ij})+1]/2 \in \{0,1\}`,
using the Binomial distribution with :math:`M=1` (a.k.a. the `Bernoulli
distribution <https://en.wikipedia.org/wiki/Bernoulli_distribution>`_),
and any of the other discrete distributions for the magnitude,
:math:`y_{ij} = \operatorname{abs}(x_{ij})`.
   
The support for weighted networks is activated by passing the parameters
``recs`` and ``rec_types`` to :class:`~graph_tool.inference.BlockState`
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(or :class:`~graph_tool.inference.OverlapBlockState`), that specify the
edge covariates (an edge :class:`~graph_tool.PropertyMap`) and their
types (a string from the table above), respectively. Note that these
parameters expect *lists*, so that multiple edge weights can be used
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simultaneously.

For example, let us consider a network of suspected terrorists involved
in the train bombing of Madrid on March 11, 2004
[hayes-connecting-2006]_. An edge indicates that a connection between
the two persons have been identified, and the weight of the edge (an
integer in the range :math:`[0,3]`) indicates the "strength" of the
connection. We can apply the weighted SBM, using a Binomial model for
the weights, as follows:


.. testsetup:: weighted-model
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   import os
   try:
       os.chdir("demos/inference")
   except FileNotFoundError:
       pass
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   gt.seed_rng(42)
         
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.. testcode:: weighted-model
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   g = gt.collection.konect_data["moreno_train"]
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   # This network contains an internal edge property map with name
   # "weight" that contains the strength of interactions. The values
   # integers in the range [0, 3].
   
   state = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[g.ep.weight],
                                                               rec_types=["discrete-binomial"]))
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   state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6),
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              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")
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.. figure:: moreno-train-wsbm.*
   :align: center
   :width: 350px

   Best fit of the Binomial-weighted degree-corrected SBM for a network
   of terror suspects, using the strength of connection as edge
   covariates. The edge colors and widths correspond to the strengths.

Model selection
+++++++++++++++

In order to select the best weighted model, we proceed in the same
manner as described in Sec. :ref:`model_selection`. However, when using
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transformations on continuous weights, we must include the associated
scaling of the probability density, as described in
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[peixoto-weighted-2017]_.

For example, consider a `food web
<https://en.wikipedia.org/wiki/Food_web>`_ between species in south
Florida [ulanowicz-network-2005]_. A directed link exists from species
:math:`i` to :math:`j` if a biomass flow exists between them, and a
weight :math:`x_{ij}` on this edge indicates the magnitude of biomass
flow (a positive real value, i.e. :math:`x_{ij}\in [0,\infty]`). One
possibility, therefore, is to use the ``"real-exponential"`` model, as
follows:
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.. testsetup:: food-web

   import os
   try:
       os.chdir("demos/inference")
   except FileNotFoundError:
       pass
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   gt.seed_rng(44)
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.. testcode:: food-web

   g = gt.collection.konect_data["foodweb-baywet"]

   # This network contains an internal edge property map with name
   # "weight" that contains the biomass flow between species. The values
   # are continuous in the range [0, infinity].
   
   state = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[g.ep.weight],
                                                               rec_types=["real-exponential"]))

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   state.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),
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              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")

.. figure:: foodweb-wsbm.*
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   :align: center
   :width: 350px

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   Best fit of the exponential-weighted degree-corrected SBM for a food
   web, using the biomass flow as edge covariates (indicated by the edge
   colors and widths).

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Alternatively, we may consider a transformation of the type
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.. math::
   :label: log_transform
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   y_{ij} = \ln x_{ij}
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so that :math:`y_{ij} \in [-\infty,\infty]`. If we use a model
``"real-normal"`` for :math:`\boldsymbol y`, it amounts to a `log-normal
<https://en.wikipedia.org/wiki/Log-normal_distribution>`_ model for
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:math:`\boldsymbol x`. This can be a better choice if the weights are
distributed across many orders of magnitude, or show multi-modality. We
can fit this alternative model simply by using the transformed weights:
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.. testcode:: food-web

   # Apply the weight transformation
   y = g.ep.weight.copy()
   y.a = log(y.a)
   
   state_ln = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[y],
                                                                  rec_types=["real-normal"]))

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                 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")

.. figure:: foodweb-wsbm-lognormal.*
   :align: center
   :width: 350px
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   Best fit of the log-normal-weighted degree-corrected SBM for a food
   web, using the biomass flow as edge covariates (indicated by the edge
   colors and widths).

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At this point, we ask ourselves which of the above models yields the
best fit of the data. This is answered by performing model selection via
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posterior odds ratios just like in Sec. :ref:`model_selection`. However,
here we need to take into account the scaling of the probability density
incurred by the variable transformation, i.e.

.. math::

    P(\boldsymbol x | \boldsymbol G, \boldsymbol b) =
    P(\boldsymbol y(\boldsymbol x) | \boldsymbol G, \boldsymbol b)
    \prod_{ij}\left[\frac{\mathrm{d}y_{ij}}{\mathrm{d}x_{ij}}(x_{ij})\right]^{A_{ij}}.

In the particular case of Eq. :eq:`log_transform`, we have

.. math::

    \prod_{ij}\left[\frac{\mathrm{d}y_{ij}}{\mathrm{d}x_{ij}}(x_{ij})\right]^{A_{ij}}
    = \prod_{ij}\frac{1}{x_{ij}^{A_{ij}}}.

Therefore, we can compute the posterior odds ratio between both models as:

.. testcode:: food-web

   L1 = -state.entropy()
   L2 = -state_ln.entropy() - log(g.ep.weight.a).sum()
              
   print(u"ln \u039b: ", L2 - L1)

.. testoutput:: food-web
   :options: +NORMALIZE_WHITESPACE

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   ln Λ:  -70.145685...
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A value of :math:`\Lambda \approx \mathrm{e}^{-70} \approx 10^{-30}` in
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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.
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Posterior sampling
++++++++++++++++++
   
The procedure to sample from the posterior distribution is identical to
what is described in Sec. :ref:`sampling`, but with the appropriate
initialization, i.e.

.. testcode:: weighted-model

    state = gt.BlockState(g, B=20, recs=[g.ep.weight], rec_types=["discrete