# Inferring network structure¶

`graph-tool`

includes algorithms to identify the large-scale structure
of networks in the `inference`

submodule. Here we
explain the basic functionality with self-contained examples.

## 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 that include the idea of “modules” in their descriptions, which then can be detected by inferring the model parameters from data. More precisely, given the partition \(\boldsymbol b = \{b_i\}\) of the network into \(B\) groups, where \(b_i\in[0,B-1]\) is the group membership of node \(i\), we define a model that generates a network \(\boldsymbol G\) with a probability

where \(\boldsymbol\theta\) are additional model parameters. Therefore, if we observe a network \(\boldsymbol G\), the likelihood that it was generated by a given partition \(\boldsymbol b\) is obtained via the Bayesian posterior

where \(P(\boldsymbol\theta, \boldsymbol b)\) is the prior likelihood of the model parameters, and

is called the model 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 \(\boldsymbol\theta\) that is compatible with the generated network, such that Eq. (2) simplifies to

with \(\boldsymbol\theta\) above being the only choice compatible with \(\boldsymbol G\) and \(\boldsymbol b\). The inference procedures considered here will consist in either finding a network partition that maximizes Eq. (4), or sampling different partitions according its posterior probability.

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

### Minimum description length (MDL)¶

We note that Eq. (4) can be written as

where

is called the **description length** of the network \(\boldsymbol G\). It
measures the amount of information required to
describe the data, if we encode it using the
particular parametrization of the generative model given by
\(\boldsymbol\theta\) and \(\boldsymbol b\), as well as the parameters
themselves. Therefore, if we choose to maximize the posterior likelihood
of Eq. (4) it will be fully equivalent to the
so-called minimum description length
method. This approach corresponds to an implementation of Occam’s 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, i.e. mistake stochastic
fluctuations for actual structure.

## The stochastic block model (SBM)¶

The stochastic block model is arguably the simplest generative process based on the notion of groups of nodes [holland-stochastic-1983]. The microcanonical formulation [peixoto-nonparametric-2016] of the basic or “traditional” version takes as parameters the partition of the nodes into groups \(\boldsymbol b\) and a \(B\times B\) matrix of edge counts \(\boldsymbol e\), where \(e_{rs}\) is the number of edges between groups \(r\) and \(s\). 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.

Note

We emphasize that no constraints are imposed on what kind of modular structure is allowed. Hence, we can detect the putatively typical pattern of “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, core-periphery, and many others, all under the same inference framework.

Although quite general, the traditional model assumes that the edges are placed randomly inside each group, and as such the nodes that belong to the same group 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 \(\boldsymbol k = \{k_i\}\) of the graph as an additional set of parameters (assuming again a microcanonical formulation [peixoto-nonparametric-2016]).

### The nested stochastic block model¶

The regular SBM has a drawback when applied to very large networks. Namely, it cannot be used to find relatively small groups in very large networks: The maximum number of groups that can be found scales as \(B_{\text{max}}\sim\sqrt{N}\), where \(N\) is the number of nodes in the network, if Bayesian inference is performed [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 \(e\) of edge counts are generated from another SBM, and so on recursively [peixoto-hierarchical-2014].

In addition to being able to find small groups in large networks, this model also provides a multilevel hierarchical description of the network, that describes its structure at multiple scales.

## Inferring the best partition¶

The simplest and most efficient approach is to find the best
partition of the network by maximizing Eq. (4)
according to some version of the model. This is obtained via the
functions `minimize_blockmodel_dl()`

or
`minimize_nested_blockmodel_dl()`

, which
employs an agglomerative multilevel Markov chain Monte Carlo (MCMC) 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
`collection`

module:

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

which yields

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

we then fit the traditional model by calling

```
state = gt.minimize_blockmodel_dl(g, deg_corr=False)
```

This returns a `BlockState`

object that
includes the inference results.

Note

The inference algorithm used is stochastic by nature, and may return a slightly different answer each time it is run. This may be due to the fact that there are alternative partitions with similar likelihoods, or that the optimum is difficult to find. Note that the inference problem here is, in general, NP-Hard, 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,
and select the partition with the largest posterior likelihood of
Eq. (4), or equivalently, the minimum description
length of Eq. (5). The description length of a fit can be
obtained with the `entropy()`

method. See also Hierarchical partitions below.

We may perform a drawing of the partition obtained via the
`draw`

method, that functions as a
convenience wrapper to the `graph_draw()`

function

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

which yields the following image.

We can obtain the group memberships as a
`PropertyMap`

on the vertices via the
`get_blocks`

method:

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

which yields:

```
3
```

We may also access the matrix of edge counts between groups via
`get_matrix`

```
e = state.get_matrix()
matshow(e.todense())
savefig("football-edge-counts.svg")
```

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:

```
bg = state.get_bg()
ers = bg.ep.count # edge counts
nr = bg.vp.count # node counts
```

### Hierarchical partitions¶

The inference of the nested family of SBMs is done in a similar manner,
but we must use instead the
`minimize_nested_blockmodel_dl()`

function. We
illustrate its use with the neural network of the C. elegans worm:

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

which has 297 vertices and 2359 edges.

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

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

```
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
```

The object returned is an instance of a
`NestedBlockState`

class, which
encapsulates the results. We can again draw the resulting hierarchical
clustering using the
`draw()`

method:

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

Note

If the `output`

parameter to
`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
`print_summary()`

method,
which shows the number of nodes and groups in all levels:

```
state.print_summary()
```

```
l: 0, N: 297, B: 13
l: 1, N: 13, B: 5
l: 2, N: 5, B: 2
l: 3, N: 2, B: 1
```

The hierarchical levels themselves are represented by individual
`BlockState()`

instances obtained via the
`get_levels()`

method:

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

```
<BlockState object with 13 blocks (13 nonempty), degree-corrected, for graph <Graph object, directed, with 297 vertices and 2359 edges at 0x...>, at 0x...>
<BlockState object with 5 blocks (5 nonempty), for graph <Graph object, directed, with 13 vertices and 105 edges at 0x...>, at 0x...>
<BlockState object with 2 blocks (2 nonempty), for graph <Graph object, directed, with 5 vertices and 21 edges at 0x...>, at 0x...>
<BlockState object with 1 blocks (1 nonempty), for graph <Graph object, directed, with 2 vertices and 4 edges at 0x...>, at 0x...>
```

This means that we can inspect the hierarchical partition just as before:

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

```
2
1
0
```

### 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

```
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())
```

```
Non-degree-corrected DL: 8507.97432099
Degree-corrected DL: 8228.11609772
```

Since it yields the smallest description length, the degree-corrected fit should be preferred. The statistical significance of the choice can be accessed by inspecting the posterior odds ratio [peixoto-nonparametric-2016]

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

```
print(u"ln Λ: ", state_dc.entropy() - state_ndc.entropy())
```

```
ln Λ: -279.858223272
```

The precise threshold that should be used to decide when to reject a hypothesis is subjective and context-dependent, but the value above implies that the particular degree-corrected fit is around \(e^{280} \sim 10^{121}\) 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:

```
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())
print(u"ln Λ:\t\t\t", state_ndc.entropy() - state_dc.entropy())
```

```
Non-degree-corrected DL: 1749.51938237
Degree-corrected DL: 1780.57671694
ln Λ: -31.0573345685
```

Hence, with a posterior odds ratio of \(\Lambda \sim e^{-36} \sim 10^{-16}\) in favor of the non-degree-corrected model, it seems like the degree-corrected variant is an unnecessarily complex description for this network.

## Averaging over models¶

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 likelihoods. In such situations, one should instead sample partitions from the posterior likelihood, instead of simply finding its maximum. One can then compute quantities that are averaged over the different model fits, weighted according to their posterior likelihoods.

Full support for model averaging is implemented in `graph-tool`

via an
efficient Markov chain Monte Carlo (MCMC) algorithm
[peixoto-efficient-2014]. It works by attempting to move nodes into
different groups with specific probabilities, and accepting or
rejecting
such moves such that, after a sufficiently long time, the partitions
will be observed with the desired posterior probability. The algorithm
is so designed, that its run-time is independent on the number of groups
being used in the model, and hence is suitable for use on very large
networks.

In order to perform such moves, one needs again to operate with
`BlockState`

or
`NestedBlockState`

instances, and calling
their `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

```
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.
# If we work with the above state object, we will be restricted to
# partitions into at most B=20 groups. But since we want to consider
# an arbitrary number of groups in the range [1, N], we transform it
# into a state with B=N groups (where N-20 will be empty).
state = state.copy(B=g.num_vertices())
# Now we run 1,000 sweeps of the MCMC
dS, nmoves = state.mcmc_sweep(niter=1000)
print("Change in description length:", dS)
print("Number of accepted vertex moves:", nmoves)
```

```
Change in description length: -355.396342...
Number of accepted vertex moves: 4561
```

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
`BlockState`

by hand. Instead, a much
better option in practice is to start from the “ground state”
obtained with `minimize_blockmodel_dl()`

,
e.g.

state = gt.minimize_blockmodel_dl(g) state = state.copy(B=g.num_vertices()) dS, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves)Change in description length: 7.34234097... Number of accepted vertex moves: 3939

Although the above is sufficient to implement model averaging, there is a
convenience function called
`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,

```
# 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:

```
niter: 1 count: 0 breaks: 0 min_S: 709.95524 max_S: 726.36140 S: 726.36140 ΔS: 16.4062 moves: 57
niter: 2 count: 1 breaks: 0 min_S: 709.95524 max_S: 726.36140 S: 721.68682 ΔS: -4.67459 moves: 67
niter: 3 count: 0 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 709.37313 ΔS: -12.3137 moves: 47
niter: 4 count: 1 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 711.61100 ΔS: 2.23787 moves: 57
niter: 5 count: 2 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 716.08147 ΔS: 4.47047 moves: 28
niter: 6 count: 3 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 712.93940 ΔS: -3.14207 moves: 47
niter: 7 count: 4 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 712.38780 ΔS: -0.551596 moves: 46
niter: 8 count: 5 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 718.00449 ΔS: 5.61668 moves: 40
niter: 9 count: 0 breaks: 0 min_S: 709.37313 max_S: 731.89940 S: 731.89940 ΔS: 13.8949 moves: 50
niter: 10 count: 0 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 707.07048 ΔS: -24.8289 moves: 45
niter: 11 count: 1 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 711.91030 ΔS: 4.83982 moves: 31
niter: 12 count: 2 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 726.56358 ΔS: 14.6533 moves: 56
niter: 13 count: 3 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 731.77165 ΔS: 5.20807 moves: 72
niter: 14 count: 4 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 707.08606 ΔS: -24.6856 moves: 57
niter: 15 count: 0 breaks: 0 min_S: 707.07048 max_S: 735.85102 S: 735.85102 ΔS: 28.7650 moves: 65
niter: 16 count: 1 breaks: 0 min_S: 707.07048 max_S: 735.85102 S: 707.29116 ΔS: -28.5599 moves: 43
niter: 17 count: 0 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 702.18860 ΔS: -5.10256 moves: 39
niter: 18 count: 1 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 716.40444 ΔS: 14.2158 moves: 55
niter: 19 count: 2 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 703.51896 ΔS: -12.8855 moves: 32
niter: 20 count: 3 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 714.30455 ΔS: 10.7856 moves: 34
niter: 21 count: 4 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 707.26722 ΔS: -7.03733 moves: 25
niter: 22 count: 5 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 730.23976 ΔS: 22.9725 moves: 21
niter: 23 count: 6 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 730.56562 ΔS: 0.325858 moves: 59
niter: 24 count: 0 breaks: 0 min_S: 702.18860 max_S: 738.45136 S: 738.45136 ΔS: 7.88574 moves: 60
niter: 25 count: 0 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 740.29015 ΔS: 1.83879 moves: 88
niter: 26 count: 1 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 720.86367 ΔS: -19.4265 moves: 68
niter: 27 count: 2 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 723.60308 ΔS: 2.73941 moves: 48
niter: 28 count: 3 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 732.81310 ΔS: 9.21002 moves: 44
niter: 29 count: 4 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 729.62283 ΔS: -3.19028 moves: 62
niter: 30 count: 5 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 730.15676 ΔS: 0.533935 moves: 59
niter: 31 count: 6 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 728.27350 ΔS: -1.88326 moves: 65
niter: 32 count: 7 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 732.19406 ΔS: 3.92056 moves: 57
niter: 33 count: 8 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 730.53906 ΔS: -1.65500 moves: 72
niter: 34 count: 9 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 725.59638 ΔS: -4.94268 moves: 72
niter: 35 count: 0 breaks: 1 min_S: 733.07687 max_S: 733.07687 S: 733.07687 ΔS: 7.48049 moves: 54
niter: 36 count: 0 breaks: 1 min_S: 728.56326 max_S: 733.07687 S: 728.56326 ΔS: -4.51361 moves: 57
niter: 37 count: 0 breaks: 1 min_S: 728.56326 max_S: 755.55140 S: 755.55140 ΔS: 26.9881 moves: 83
niter: 38 count: 0 breaks: 1 min_S: 728.56326 max_S: 761.09434 S: 761.09434 ΔS: 5.54294 moves: 96
niter: 39 count: 0 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 713.60740 ΔS: -47.4869 moves: 71
niter: 40 count: 1 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 713.98904 ΔS: 0.381637 moves: 67
niter: 41 count: 2 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 729.22460 ΔS: 15.2356 moves: 68
niter: 42 count: 3 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 724.70143 ΔS: -4.52317 moves: 69
niter: 43 count: 0 breaks: 1 min_S: 703.51896 max_S: 761.09434 S: 703.51896 ΔS: -21.1825 moves: 40
niter: 44 count: 0 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 702.85027 ΔS: -0.668696 moves: 33
niter: 45 count: 1 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 722.46508 ΔS: 19.6148 moves: 49
niter: 46 count: 2 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 714.77930 ΔS: -7.68578 moves: 62
niter: 47 count: 3 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 722.04551 ΔS: 7.26621 moves: 55
niter: 48 count: 4 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 708.96879 ΔS: -13.0767 moves: 37
niter: 49 count: 5 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 714.84009 ΔS: 5.87130 moves: 37
niter: 50 count: 6 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 718.28558 ΔS: 3.44549 moves: 55
niter: 51 count: 7 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 720.86398 ΔS: 2.57840 moves: 44
niter: 52 count: 8 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 710.93672 ΔS: -9.92726 moves: 45
niter: 53 count: 9 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 735.06773 ΔS: 24.1310 moves: 28
niter: 54 count: 10 breaks: 2 min_S: 702.85027 max_S: 761.09434 S: 738.16756 ΔS: 3.09983 moves: 115
```

Note that the value of `wait`

above was made purposefully low so that
the output would not be overly long. The most appropriate value requires
experimentation, but a typically good value is `wait=1000`

.

The function `mcmc_equilibrate()`

accepts a
`callback`

argument that takes an optional function to be invoked
after each call to
`mcmc_sweep()`

. This function
should accept a single parameter which will contain the actual
`BlockState`

instance. We will use this in
the example below to collect the posterior vertex marginals, i.e. the
posterior probability that a node belongs to a given group:

```
# 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)
# 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 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")
```

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

```
h = np.zeros(g.num_vertices() + 1)
def collect_num_groups(s):
B = s.get_nonempty_B()
h[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)
```

### 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
`NestedBlockState`

instances.

Note

When using `NestedBlockState`

instances
to perform model averaging, they need to be constructed with the
option `sampling=True`

.

Here we perform the sampling of hierarchical partitions using the same network as above.

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

```
Change in description length: 6.222068...
Number of accepted vertex moves: 7615
```

Similarly to the the non-nested case, we can use
`mcmc_equilibrate()`

to do most of the boring
work, and we can now obtain vertex marginals on all hierarchical levels:

```
# 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")
```

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

```
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)
```

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

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

### Model class selection¶

When averaging over partitions, we may be interested in evaluating which
**model class** provides a better fit of the data, considering all
possible parameter choices. This is done by evaluating the model
evidence [peixoto-nonparametric-2016]

This quantity is analogous to a partition function in statistical physics, which we can write more conveniently as a negative free energy by taking its logarithm

where

is the posterior likelihood of partition \(\boldsymbol b\). The first term of Eq. (6) (the “negative energy”) is minus the average of description length \(\left<\Sigma\right>\), weighted according to the posterior distribution. The second term \(\mathcal{S}\) is the entropy 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 single partition with a very large likelihood, the entropy will tend to zero. However, if there are many partitions with similar likelihoods — meaning that there is no single partition that describes the network uniquely well — it will take a large value instead.

Since the MCMC algorithm samples partitions from the distribution \(q(\boldsymbol b)\), it can be used to compute \(\left<\Sigma\right>\) easily, simply by averaging the description length values encountered by sampling from the posterior distribution many times.

The computation of the posterior entropy \(\mathcal{S}\), however,
is significantly more difficult, since it involves measuring the precise
value of \(q(\boldsymbol b)\). A direct “brute force” computation of
\(\mathcal{S}\) is implemented via
`collect_partition_histogram()`

and
`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

where

is the marginal group membership distribution of node \(i\). This yields an entropy value given by

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.

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

where \(A_{ij}\) is the adjacency matrix, \(k_i\) is the degree of node \(i\), and

is the joint group membership distribution of nodes \(i\) and \(j\) (a.k.a. the edge marginals). This yields an entropy value given by

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
`collect_vertex_marginals()`

,
`collect_edge_marginals()`

,
`mf_entropy()`

and
`bethe_entropy()`

.

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

```
Model evidence for deg_corr = True: -575.864972067 (mean field), -802.39062289 (Bethe)
Model evidence for deg_corr = False: -584.307313493 (mean field), -707.827204203 (Bethe)
```

If we consider the more accurate approximation, the outcome shows a preference for the non-degree-corrected model.

When using the nested model, the approach is entirely analogous. The only difference now is that we have a hierarchical partition \(\{\boldsymbol b_l\}\) in the equations above, instead of simply \(\boldsymbol b\). In order to make the approach tractable, we assume the factorization

where \(q_l(\boldsymbol b_l)\) is the marginal posterior for the partition at level \(l\). For \(q_0(\boldsymbol b_0)\) we may use again either the mean-field or Bethe approximations, however for \(l>0\) 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.

```
g = gt.collection.data["lesmis"]
L = 10
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.
bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with
# 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())
# Now we collect the marginal distributions for exactly 200,000 sweeps
gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10),
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)")
```

```
Model evidence for deg_corr = True: -346.618790006 (mean field), -601.313781849 (Bethe)
Model evidence for deg_corr = False: -374.614350884 (mean field), -563.256840699 (Bethe)
```

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.

## Edge layers and covariates¶

In many situations, the edges of the network may posses discrete
covariates on them, or they may be distributed in discrete
“layers”. Extensions to the SBM may be defined for such data, and they
can be inferred using the exact same interface shown above, except one
should use the `LayeredBlockState`

class,
instead of `BlockState`

. This class takes
two additional parameters: the `ec`

parameter, that must correspond to
an edge `PropertyMap`

with the layer/covariate
values on the edges, and the Boolean `layers`

parameter, which if
`True`

specifies a layered model, otherwise one with edge covariates.

If we use `minimize_blockmodel_dl()`

, this can
be achieved simply by passing the option `layers=True`

as well as the
appropriate value of `state_args`

, which will be propagated to
`LayeredBlockState`

‘s constructor.

For example, consider again the Les Misérables network, where we consider the number of co-appearances between characters as edge covariates.

```
g = gt.collection.data["lesmis"]
# Note the different meaning of the two 'layers' parameters below: The
# first enables the use of LayeredBlockState, and the second selects
# the 'edge covariates' version.
state = gt.minimize_blockmodel_dl(g, deg_corr=False, layers=True,
state_args=dict(ec=g.ep.value, layers=False))
state.draw(pos=g.vp.pos, edge_color=g.ep.value, edge_gradient=None,
output="lesmis-sbm-edge-cov.svg")
```

In the case of the nested model, we still should use the
`NestedBlockState`

class, but it must be
initialized with the parameter `base_type = LayeredBlockState`

. But if
we use `minimize_nested_blockmodel_dl()`

, it
works identically to the above:

```
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=False, layers=True,
state_args=dict(ec=g.ep.value, layers=False))
state.draw(eprops=dict(color=g.ep.value, gradient=None),
output="lesmis-nested-sbm-edge-cov.svg")
```

It is possible to perform model averaging of all layered variants exactly like for the regular SBMs as was shown above.

## Predicting spurious and missing edges¶

An important application of generative models is to be able to generalize from observations and make predictions that go beyond what is seen in the data. This is particularly useful when the network we observe is incomplete, or contains errors, i.e. some of the edges are either missing or are outcomes of mistakes in measurement. In this situation, the fit we make of the observed network can help us predict missing or spurious edges in the network [clauset-hierarchical-2008] [guimera-missing-2009].

We do so by dividing the edges into two sets \(\boldsymbol G\) and \(\delta \boldsymbol G\), where the former corresponds to the observed network and the latter either to the missing or spurious edges. In the case of missing edges, we may compute the posterior of \(\delta \boldsymbol G\) as

where

is a normalization constant. Although the value of \(P_{\delta}(\boldsymbol G)\) is difficult to obtain in general (since we need to perform a sum over all possible spurious/missing edges), the numerator of Eq. (7) can be computed by sampling partitions from the posterior, and then inserting or deleting edges from the graph and computing the new likelihood. This means that we can easily compare alternative predictive hypotheses \(\{\delta \boldsymbol G_i\}\) via their likelihood ratios

which do not depend on the value of \(P_{\delta}(\boldsymbol G)\).

The values \(P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)\)
can be computed with
`get_edges_prob()`

. Hence, we can
compute spurious/missing edge probabilities just as if we were
collecting marginal distributions when doing model averaging.

Below is an example for predicting the two following edges in the football network, using the nested model (for which we need to replace \(\boldsymbol b\) by \(\{\boldsymbol b_l\}\) in the equations above).

```
g = gt.collection.data["football"]
missing_edges = [(101, 102), (17, 56)]
L = 10
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
bs = state.get_bs() # Get hierarchical partition.
bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with
# single-group levels.
state = state.copy(bs=bs, sampling=True)
probs = ([], [])
def collect_edge_probs(s):
p1 = s.get_edges_prob([missing_edges[0]], entropy_args=dict(partition_dl=False))
p2 = s.get_edges_prob([missing_edges[1]], entropy_args=dict(partition_dl=False))
probs[0].append(p1)
probs[1].append(p2)
# Now we collect the probabilities for exactly 10,000 sweeps
gt.mcmc_equilibrate(state, force_niter=1000, mcmc_args=dict(niter=10),
callback=collect_edge_probs)
def get_avg(p):
p = np.array(p)
pmax = p.max()
p -= pmax
return pmax + log(exp(p).mean())
p1 = get_avg(probs[0])
p2 = get_avg(probs[1])
p_sum = get_avg([p1, p2]) + log(2)
l1 = p1 - p_sum
l2 = p2 - p_sum
print("likelihood-ratio for %s: %g" % (missing_edges[0], exp(l1)))
print("likelihood-ratio for %s: %g" % (missing_edges[1], exp(l2)))
```

```
likelihood-ratio for (101, 102): 0.372308
likelihood-ratio for (17, 56): 0.627692
```

From which we can conclude that edge \((17, 56)\) is around twice as likely as \((101, 102)\) to be a missing edge.

The prediction using the non-nested model can be performed in an entirely analogous fashion.

## References¶

[holland-stochastic-1983] | Paul W. Holland, Kathryn Blackmond Laskey, Samuel Leinhardt, “Stochastic blockmodels: First steps”, Social Networks Volume 5, Issue 2, Pages 109-137 (1983), DOI: 10.1016/0378-8733(83)90021-7 |

[karrer-stochastic-2011] | Brian Karrer, M. E. J. Newman “Stochastic blockmodels and community structure in networks”, Phys. Rev. E 83, 016107 (2011), DOI: 10.1103/PhysRevE.83.016107, arXiv: 1008.3926 |

[peixoto-nonparametric-2016] | (1, 2, 3, 4) Tiago P. Peixoto, “Nonparametric
Bayesian inference of the microcanonical stochastic block model”
arXiv: 1610.02703 |

[peixoto-parsimonious-2013] | Tiago P. Peixoto, “Parsimonious module inference in large networks”, Phys. Rev. Lett. 110, 148701 (2013), DOI: 10.1103/PhysRevLett.110.148701, arXiv: 1212.4794. |

[peixoto-hierarchical-2014] | Tiago P. Peixoto, “Hierarchical block structures and high-resolution model selection in large networks”, Phys. Rev. X 4, 011047 (2014), DOI: 10.1103/PhysRevX.4.011047, arXiv: 1310.4377. |

[peixoto-model-2016] | Tiago P. Peixoto, “Model selection and hypothesis testing for large-scale network models with overlapping groups”, Phys. Rev. X 5, 011033 (2016), DOI: 10.1103/PhysRevX.5.011033, arXiv: 1409.3059. |

[peixoto-efficient-2014] | (1, 2) Tiago P. Peixoto, “Efficient Monte Carlo and
greedy heuristic for the inference of stochastic block models”, Phys.
Rev. E 89, 012804 (2014), DOI: 10.1103/PhysRevE.89.012804,
arXiv: 1310.4378 |

[clauset-hierarchical-2008] | Aaron Clauset, Cristopher Moore, M. E. J. Newman, “Hierarchical structure and the prediction of missing links in networks”, Nature 453, 98-101 (2008), DOI: 10.1038/nature06830 |

[guimera-missing-2009] | Roger Guimerà, Marta Sales-Pardo, “Missing and spurious interactions and the reconstruction of complex networks”, PNAS vol. 106 no. 52 (2009), DOI: 10.1073/pnas.0908366106 |

[mezard-information-2009] | Marc Mézard, Andrea Montanari, “Information, Physics, and Computation”, Oxford Univ Press, 2009. DOI: 10.1093/acprof:oso/9780198570837.001.0001 |