Inferring modular 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. For a more thorough theoretical introduction to the methods described here, the reader is referred to [peixoto-bayesian-2017].

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

(1)\[P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)\]

where \(\boldsymbol\theta\) are additional model parameters that control how the node partition affects the structure of the network. 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

(2)\[P(\boldsymbol b | \boldsymbol G) = \frac{\sum_{\boldsymbol\theta}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}\]

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

(3)\[P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)\]

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

(4)\[P(\boldsymbol b | \boldsymbol G) = \frac{P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}\]

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 also enables 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

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

where

(5)\[\Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b)\]

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 distribution 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. 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].

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-2017] 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.

../../_images/sbm-example-ers.svg

Matrix of edge counts \(\boldsymbol e\) between groups.

../../_images/sbm-example.svg

Generated network.

Note

We emphasize that no constraints are imposed on what kind of modular structure is allowed, as the matrix of edge counts \(e\) is unconstrained. 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 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 \(\boldsymbol k = \{k_i\}\) of the graph as an additional set of parameters (assuming again a microcanonical formulation [peixoto-nonparametric-2017]).

The nested stochastic block model

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 \(B_{\text{max}}=O(\sqrt{N})\), 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], as illustrated below.

../../_images/nested-diagram.svg

Example of a nested SBM with three levels.

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

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 degree-corrected model by calling

state = gt.minimize_blockmodel_dl(g)

This returns a BlockState object that includes the inference results.

Note

The inference algorithm used is stochastic by nature, and may return 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, 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 probability 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 Sec. 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.

../../_images/football-sbm-fit.svg

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 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")
../../_images/football-edge-counts.svg

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:

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

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")
../../_images/celegans-hsbm-fit.svg

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 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: 14
l: 1, N: 14, 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 14 blocks (14 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 14 vertices and 129 edges at 0x...>, at 0x...>
<BlockState object with 2 blocks (2 nonempty), for graph <Graph object, directed, with 5 vertices and 22 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:      8511.005312...
Degree-corrected DL:  8225.167736...

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-2017]

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

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 \u039b: ", state_dc.entropy() - state_ndc.entropy())
ln Λ:  -285.837575...

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 \(\mathrm{e}^{327} \approx 10^{142}\) 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 \u039b:\t\t\t", state_ndc.entropy() - state_dc.entropy())
Non-degree-corrected DL:      1733.525685...
Degree-corrected DL:          1791.750418...
ln Λ:                         -58.224733...

Hence, with a posterior odds ratio of \(\Lambda \approx \mathrm{e}^{-58} \approx 10^{-26}\) in favor of the non-degree-corrected model, it seems like the degree-corrected variant is an unnecessarily complex description for this network.

Sampling from the posterior distribution

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 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.

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 so that, after a sufficiently long time, the partitions will 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.

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.

# 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.

dS, nmoves = state.mcmc_sweep(niter=1000)

print("Change in description length:", dS)
print("Number of accepted vertex moves:", nmoves)
Change in description length: -345.376523...
Number of accepted vertex moves: 34222

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 an approximation to 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: 16.124022...
Number of accepted vertex moves: 41393

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: 713.72081  max_S: 720.04438  S: 713.72081  ΔS:     -6.32357  moves:   418
niter:     2  count:    0  breaks:  0  min_S: 713.72081  max_S: 728.31214  S: 728.31214  ΔS:      14.5913  moves:   384
niter:     3  count:    0  breaks:  0  min_S: 713.70119  max_S: 728.31214  S: 713.70119  ΔS:     -14.6110  moves:   443
niter:     4  count:    1  breaks:  0  min_S: 713.70119  max_S: 728.31214  S: 722.48803  ΔS:      8.78684  moves:   391
niter:     5  count:    2  breaks:  0  min_S: 713.70119  max_S: 728.31214  S: 727.05935  ΔS:      4.57131  moves:   378
niter:     6  count:    3  breaks:  0  min_S: 713.70119  max_S: 728.31214  S: 727.29821  ΔS:     0.238862  moves:   344
niter:     7  count:    0  breaks:  0  min_S: 713.70119  max_S: 743.00358  S: 743.00358  ΔS:      15.7054  moves:   376
niter:     8  count:    0  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 711.80965  ΔS:     -31.1939  moves:   382
niter:     9  count:    1  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 712.92615  ΔS:      1.11651  moves:   343
niter:    10  count:    2  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 721.94043  ΔS:      9.01428  moves:   388
niter:    11  count:    3  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 719.13006  ΔS:     -2.81037  moves:   362
niter:    12  count:    4  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 729.78095  ΔS:      10.6509  moves:   383
niter:    13  count:    5  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 720.04992  ΔS:     -9.73104  moves:   376
niter:    14  count:    6  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 732.90657  ΔS:      12.8567  moves:   387
niter:    15  count:    7  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 717.42580  ΔS:     -15.4808  moves:   380
niter:    16  count:    8  breaks:  0  min_S: 711.80965  max_S: 743.00358  S: 716.75399  ΔS:    -0.671812  moves:   359
niter:    17  count:    0  breaks:  0  min_S: 711.80965  max_S: 745.15972  S: 745.15972  ΔS:      28.4057  moves:   350
niter:    18  count:    1  breaks:  0  min_S: 711.80965  max_S: 745.15972  S: 728.99832  ΔS:     -16.1614  moves:   389
niter:    19  count:    2  breaks:  0  min_S: 711.80965  max_S: 745.15972  S: 720.84596  ΔS:     -8.15237  moves:   348
niter:    20  count:    0  breaks:  0  min_S: 709.75049  max_S: 745.15972  S: 709.75049  ΔS:     -11.0955  moves:   392
niter:    21  count:    1  breaks:  0  min_S: 709.75049  max_S: 745.15972  S: 721.10373  ΔS:      11.3532  moves:   341
niter:    22  count:    2  breaks:  0  min_S: 709.75049  max_S: 745.15972  S: 718.50836  ΔS:     -2.59537  moves:   354
niter:    23  count:    3  breaks:  0  min_S: 709.75049  max_S: 745.15972  S: 714.36017  ΔS:     -4.14819  moves:   375
niter:    24  count:    0  breaks:  0  min_S: 707.10762  max_S: 745.15972  S: 707.10762  ΔS:     -7.25255  moves:   367
niter:    25  count:    1  breaks:  0  min_S: 707.10762  max_S: 745.15972  S: 708.42197  ΔS:      1.31435  moves:   372
niter:    26  count:    0  breaks:  0  min_S: 704.56635  max_S: 745.15972  S: 704.56635  ΔS:     -3.85562  moves:   346
niter:    27  count:    1  breaks:  0  min_S: 704.56635  max_S: 745.15972  S: 725.76740  ΔS:      21.2011  moves:   338
niter:    28  count:    2  breaks:  0  min_S: 704.56635  max_S: 745.15972  S: 708.78787  ΔS:     -16.9795  moves:   378
niter:    29  count:    3  breaks:  0  min_S: 704.56635  max_S: 745.15972  S: 722.03356  ΔS:      13.2457  moves:   382
niter:    30  count:    0  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 704.10199  ΔS:     -17.9316  moves:   375
niter:    31  count:    1  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 713.64366  ΔS:      9.54166  moves:   382
niter:    32  count:    2  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 727.65050  ΔS:      14.0068  moves:   383
niter:    33  count:    3  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 720.51443  ΔS:     -7.13607  moves:   379
niter:    34  count:    4  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 726.77412  ΔS:      6.25969  moves:   366
niter:    35  count:    5  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 722.96778  ΔS:     -3.80634  moves:   382
niter:    36  count:    6  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 717.65450  ΔS:     -5.31328  moves:   394
niter:    37  count:    7  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 734.02750  ΔS:      16.3730  moves:   377
niter:    38  count:    8  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 727.36795  ΔS:     -6.65954  moves:   393
niter:    39  count:    9  breaks:  0  min_S: 704.10199  max_S: 745.15972  S: 719.39773  ΔS:     -7.97022  moves:   382
niter:    40  count:    0  breaks:  1  min_S: 718.06061  max_S: 718.06061  S: 718.06061  ΔS:     -1.33712  moves:   411
niter:    41  count:    0  breaks:  1  min_S: 708.36787  max_S: 718.06061  S: 708.36787  ΔS:     -9.69274  moves:   398
niter:    42  count:    0  breaks:  1  min_S: 708.36787  max_S: 729.98460  S: 729.98460  ΔS:      21.6167  moves:   406
niter:    43  count:    1  breaks:  1  min_S: 708.36787  max_S: 729.98460  S: 719.27340  ΔS:     -10.7112  moves:   383
niter:    44  count:    2  breaks:  1  min_S: 708.36787  max_S: 729.98460  S: 709.89100  ΔS:     -9.38239  moves:   409
niter:    45  count:    3  breaks:  1  min_S: 708.36787  max_S: 729.98460  S: 721.29921  ΔS:      11.4082  moves:   383
niter:    46  count:    0  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 706.67224  ΔS:     -14.6270  moves:   405
niter:    47  count:    1  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 711.87311  ΔS:      5.20087  moves:   373
niter:    48  count:    2  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 708.20851  ΔS:     -3.66460  moves:   367
niter:    49  count:    3  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 712.26954  ΔS:      4.06103  moves:   368
niter:    50  count:    4  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 717.69181  ΔS:      5.42227  moves:   396
niter:    51  count:    5  breaks:  1  min_S: 706.67224  max_S: 729.98460  S: 716.64174  ΔS:     -1.05008  moves:   395
niter:    52  count:    0  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 731.96439  ΔS:      15.3226  moves:   387
niter:    53  count:    1  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 722.51613  ΔS:     -9.44825  moves:   411
niter:    54  count:    2  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 719.18164  ΔS:     -3.33449  moves:   414
niter:    55  count:    3  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 712.43942  ΔS:     -6.74222  moves:   395
niter:    56  count:    4  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 720.71508  ΔS:      8.27565  moves:   395
niter:    57  count:    5  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 718.75450  ΔS:     -1.96058  moves:   379
niter:    58  count:    6  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 710.43596  ΔS:     -8.31854  moves:   428
niter:    59  count:    7  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 723.89819  ΔS:      13.4622  moves:   408
niter:    60  count:    8  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 718.87456  ΔS:     -5.02363  moves:   435
niter:    61  count:    9  breaks:  1  min_S: 706.67224  max_S: 731.96439  S: 721.20227  ΔS:      2.32772  moves:   399
niter:    62  count:   10  breaks:  2  min_S: 706.67224  max_S: 731.96439  S: 726.81344  ΔS:      5.61116  moves:   383

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 (via collect_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, at
# intervals of 10 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")
../../_images/lesmis-sbm-marginals.svg

Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the degree-corrected SBM. The pie fractions 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.

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

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

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

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.

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: 23.368680...
Number of accepted vertex moves: 46167

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")
../../_images/lesmis-nested-sbm-marginals.svg

Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the nested degree-corrected SBM. The pie fractions on the nodes correspond to the probability of being in group associated with the respective color.

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)
../../_images/lesmis-nested-B-posterior.svg

Marginal posterior probability of the number of nonempty groups \(B_l\) at each hierarchy level \(l\) 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.

for i in range(10):
    state.mcmc_sweep(niter=1000)
    state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False)
../../_images/lesmis-partition-sample-0.svg ../../_images/lesmis-partition-sample-1.svg ../../_images/lesmis-partition-sample-2.svg ../../_images/lesmis-partition-sample-3.svg ../../_images/lesmis-partition-sample-4.svg ../../_images/lesmis-partition-sample-5.svg ../../_images/lesmis-partition-sample-6.svg ../../_images/lesmis-partition-sample-7.svg ../../_images/lesmis-partition-sample-8.svg ../../_images/lesmis-partition-sample-9.svg

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 summed over all possible partitions [peixoto-nonparametric-2017]:

\[P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G,\boldsymbol\theta, \boldsymbol b) = \sum_{\boldsymbol b}P(\boldsymbol G,\boldsymbol b).\]

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

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

where

\[q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}\]

is the posterior probability 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 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.

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

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

where

\[q_i(r) = P(b_i = r | \boldsymbol G)\]

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

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

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.

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

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

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

\[q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)\]

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

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

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: -588.9373... (mean field), -749.3004... (Bethe)
Model evidence for deg_corr = False: -593.393... (mean field), -709.6448... (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

\[q(\{\boldsymbol b_l\}) \approx \prod_lq_l(\boldsymbol b_l)\]

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"]

nL = 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)] * (nL - 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: -554.46079... (mean field), -699.6147... (Bethe)
Model evidence for deg_corr = False: -548.6576... (mean field), -635.5347... (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 weights and covariates

Very often networks cannot be completely represented by simple graphs, but instead have arbitrary “weights” \(x_{ij}\) 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.

\[P(\boldsymbol x,\boldsymbol G|\boldsymbol b) = P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b),\]

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

\[P(\boldsymbol x|\boldsymbol G,\boldsymbol b) = \prod_{r\le s}\int P({\boldsymbol x}_{rs}|\gamma)P(\gamma)\,\mathrm{d}\gamma,\]

where \(P({\boldsymbol x}_{rs}|\gamma)\) is some model for the weights \({\boldsymbol x}_{rs}\) between groups \((r,s)\), conditioned on some parameter \(\gamma\), sampled from its prior \(P(\gamma)\). 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

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

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:

Name Domain Bounds Shape
"real-exponential" Real \((\mathbb{R})\) \([0,\infty]\) Exponential
"real-normal" Real \((\mathbb{R})\) \([-\infty,\infty]\) Normal
"discrete-geometric" Natural \((\mathbb{N})\) \([0,\infty]\) Geometric
"discrete-binomial" Natural \((\mathbb{N})\) \([0,M]\) Binomial
"discrete-poisson" Natural \((\mathbb{N})\) \([0,\infty]\) Poisson

In fact, the actual model implements microcanonical 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 weight model for positive real weights \(\boldsymbol x\), we can use the transformation \(y_{ij} = \ln x_{ij}\) together with the "real-normal" model for \(\boldsymbol y\). To model weights that are positive or negative integers in \(\mathbb{Z}\), we could either subtract the minimum value, \(y_{ij} = x_{ij} - x^*\), with \(x^*=\operatorname{min}_{ij}x_{ij}\), and use any of the above models for non-negative integers in \(\mathbb{N}\), or alternatively, consider the sign as an additional covariate, i.e. \(s_{ij} = [\operatorname{sign}(x_{ij})+1]/2 \in \{0,1\}\), using the Binomial distribution with \(M=1\) (a.k.a. the Bernoulli distribution), and any of the other discrete distributions for the magnitude, \(y_{ij} = \operatorname{abs}(x_{ij})\).

The support for weighted networks is activated by passing the parameters recs and rec_types to BlockState (or OverlapBlockState), that specify the edge covariates (an edge 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 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 \([0,3]\)) indicates the “strength” of the connection. We can apply the weighted SBM, using a Binomial model for the weights, as follows:

g = gt.collection.konect_data["moreno_train"]

# 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"]))

state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6),
           eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1),
           edge_gradient=[], output="moreno-train-wsbm.svg")
../../_images/moreno-train-wsbm.svg

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. Model selection. However, when using transformations on continuous weights, we must include the associated scaling of the probability density, as described in [peixoto-weighted-2017].

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

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"]))

state.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),
           eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True),
           edge_gradient=[], output="foodweb-wsbm.svg")
../../_images/foodweb-wsbm.svg

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).

Alternatively, we may consider a transformation of the type

(7)\[y_{ij} = \ln x_{ij}\]

so that \(y_{ij} \in [-\infty,\infty]\). If we use a model "real-normal" for \(\boldsymbol y\), it amounts to a log-normal model for \(\boldsymbol x\). 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:

# 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"]))

state_ln.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),
              eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True),
              edge_gradient=[], output="foodweb-wsbm-lognormal.svg")
../../_images/foodweb-wsbm-lognormal.svg

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).

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 posterior odds ratios just like in Sec. Model selection. However, here we need to take into account the scaling of the probability density incurred by the variable transformation, i.e.

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

In the particular case of Eq. (7), we have

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

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

L1 = -state.entropy()
L2 = -state_ln.entropy() - log(g.ep.weight.a).sum()

print(u"ln \u039b: ", L2 - L1)
ln Λ:  -43.189790...

A value of \(\Lambda \approx \mathrm{e}^{-43} \approx 10^{-19}\) in favor the exponential model indicates that the log-normal model does not provide a better fit for this particular data. Based on this, we conclude that the exponential model should be preferred in this case.

Posterior sampling

The procedure to sample from the posterior distribution is identical to what is described in Sec. Sampling from the posterior distribution, but with the appropriate initialization, i.e.

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

or for the nested model

state = gt.NestedBlockState(g, bs=[np.random.randint(0, 20, g.num_vertices())] + [zeros(1)] * 10,
                            state_args=dict(recs=[g.ep.weight],
                                            rec_types=["discrete-poisson"]))

Layered networks

The edges of the network may be distributed in discrete “layers”, representing distinct types if interactions [peixoto-inferring-2015]. 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 categorical edge covariates (not to be confused with the weighted models in Sec. Edge weights and 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.

As an example, let us consider a social network of tribes, where two types of interactions were recorded, amounting to either friendship or enmity [read-cultures-1954]. We may apply the layered model by separating these two types of interactions in two layers:

g = gt.collection.konect_data["ucidata-gama"]

# The edge types are stored in the edge property map "weights".

# Note the different meanings of the two 'layers' parameters below: The
# first enables the use of LayeredBlockState, and the second selects
# the 'edge layers' version (instead of 'edge covariates').

state = gt.minimize_nested_blockmodel_dl(g, layers=True,
                                         state_args=dict(ec=g.ep.weight, layers=True))

state.draw(edge_color=g.ep.weight, edge_gradient=[],
           ecmap=(matplotlib.cm.coolwarm_r, .6), edge_pen_width=5,
           output="tribes-sbm-edge-layers.svg")
../../_images/tribes-sbm-edge-layers.svg

Best fit of the degree-corrected SBM with edge layers for a network of tribes, with edge layers shown as colors. The groups show two enemy tribes.

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. We may compute the posterior of \(\delta \boldsymbol G\) as [valles-catala-consistency-2017]

(8)\[P(\delta \boldsymbol G | \boldsymbol G) \propto \sum_{\boldsymbol b}\frac{P(\boldsymbol G \cup \delta\boldsymbol G| \boldsymbol b)}{P(\boldsymbol G| \boldsymbol b)}P(\boldsymbol b | \boldsymbol G)\]

up to a normalization constant. Although the normalization constant is difficult to obtain in general (since we need to perform a sum over all possible spurious/missing edges), the numerator of Eq. (8) 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

\[\lambda_i = \frac{P(\delta \boldsymbol G_i | \boldsymbol G)}{\sum_j P(\delta \boldsymbol G_j | \boldsymbol G)}\]

which do not depend on the normalization constant.

The values \(P(\delta \boldsymbol G | \boldsymbol G, \boldsymbol b)\) 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).

../../_images/football_missing.svg

Two non-existing edges in the football network (in red): \((101,102)\) in the middle, and \((17,56)\) in the upper right region of the figure.

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 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, 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): 0.36...
likelihood-ratio for (17, 56): 0.63...

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

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

References

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