Author: team großBM
Date: 2024-01-31

# Stochastic Block Models for Multiplex networks

### Principle and illustration

#### team großBM

#### 2024-01-31

## Preliminaries

This vignette illustrates the use of the `estimateMultiplexSBM` function and the methods accompanying the R6 classes `multiplexSBMfit`.

### Requirements

The only package required for the analysis is sbm.

    library(sbm)

## Multiplex network data

Multiplex networks refer to a collection of networks involving the same sets of nodes, each network corresponding to a given type of interaction. Each network may also be referred as a layer. Such a network is studied in Barbillon et al. (2017) where the nodes are researchers and two networks are provided:

    * a network representing the relations of advices between researchers,
    * a network representing the exchanges between the laboratories the researchers belong to.

In Kéfi (2016), the nodes are species and each network corresponds to a type of ecological relation, namely trophic links, negative non trophic links and positive non trophic links.

## Stochastic Block models for multiplex networks

### General formulation of the model

We give here a few details on Stochastic Block Models (SBM) for multiplex networks.

Assume that one studies \(n\) nodes and observes \(L\) types of links, resulting into \(L\) matrices \((X^\ell_{ij})_{i,j = 1,\dots,n}\), \(\ell = 1, \dots, L\). Block models assume that the nodes are divided into \(K\) clusters, this clustering being encoded into the latent variables \((Z_i)_{i=1,\dots,n}\) such that \(Z_i = k\) if node \(i\) belongs to cluster \(k\).

The relations between nodes \(i\) and \(j\) are driven by the clusters they belong to :

\[ (X^1_{ij}, \dots,X^L_{ij}) | Z_i = q, Z_j = r \sim_{ind} \mathcal{F}(\cdot;\theta_{qr})\] where \(\mathcal{F}\) is a \(L\) dimensional probability distribution adapted to the data.

Examples

- If all the networks are binary (\(X^\ell_{ij} \in \{0,1\}\)) then \(\mathcal{F}\) is a \(L\)-variate Bernoulli distribution and \[ \theta_{qr} = (p^{\omega}_{qr}, \omega \in \{0,1\}^L)\] with \(p^{\omega_\ell}_{qr} \in [0,1]\).
- If all the networks are weighted with real values (\(X^\ell_{ij} \in \mathbb{R}\)) then \(\mathcal{F}\) could be a \(L\)-dimensional Gaussian distribution and \[ \theta_{qr} = (\mu_{qr},\Sigma_{qr})\] with \(\mu_{qr} \in \mathbb{R}^L\), \(\Sigma_{qr}\) a \(L\) covariance matrix.

Note that these two models assume that the \(L\) interactions are dependent conditionally to the clustering. The integration against the latent variables \(Z\) (clustering) implies that the interactions between two pairs of nodes are also dependent.

A collection of models are implemented in our package `sbm`. These models can be classified into two groups : either the layers are dependent conditionally to \(Z\) or not.

### Dependent and independent layers conditionally to \(Z\)

We consider conditional dependence in a few numbers of models :

- for \(L=2\) Binary networks (Bernoulli)

- For any number \(L\) of layers with Gaussian multivariate distributions but restricted to \(\Sigma_{qr} = \Sigma\) (same covariance in any blocks).

In the Binary case, the number of parameters of dependent layers explodes with the number of layers \(L\) and the results become difficult to understand. As a consequence, we also implemented inference methods for independent layers conditionally to the clusterings for any number of layers.

In the case of independent layers conditionally to the clusters, we assume that for any \(\ell,i,j\), \[ X^\ell_{ij} | Z_i = q, Z_j = r \sim_{ind} \mathcal{F}_\ell(\cdot;\theta^\ell_{qr})\] where any \(\mathcal{F}_\ell\) can be Bernoulli, Poisson, Gaussian or Zero Inflated Gaussian.

### Bipartite multiplex networks

All the previous models can be extended to bipartite networks (i.e. when the nodes in row are different from the nodes in column). In that case, two clusterings are introduced. Assume that the row (respectively column) nodes are divided into \(K_1\) (resp. \(K_2\)) clusters, and denote \((Z^1_i)_{i=1,\dots,n}\) (resp. \((Z^2_j)_{i=j,\dots,p}\) ) the row (resp. col) clusterings.

The relations between nodes \(i\) and \(j\) are driven by the clusters they belong to :

\[ (X^1_{ij}, \dots,X^L_{ij}) | Z^1_i = q, Z^2_j = r \sim_{ind} \mathcal{F}(\cdot;\theta_{qr})\]

### Inference

The maximization of the likelihood is obtained via a Variational version of the Expectation- Maximization algorithm. The number of blocks is chosen via a penalized likelihood criterion (ICL).
See Barbillon et al. (2017) for details.

Note that, in the case of dependent networks, the inference is performed via the `blockmodels` package, while the `GREMLINS` package is used in the case of independent networks. As a consequence, the `estimOptions` arguments are slightly different.

## Implementation

### Data simulation

The function `mySampleMultiplexSBM` supplies a method to simulate multiplex networks. The argument `dependent` is set to `FALSE` by default (assuming independent layers conditionally to the clusterings).

We simulate hereafter \(2\) simple (i.e. non bipartite) layers. The corresponding matrices can be plotted using the function `plotMyMultiplexMatrix`.

    nbLayers <- 2

Examples

- 2 directed Layers : one Poisson, one Bernoulli and two clusters

    Nnodes <- 40
    blockProp <- c(.4,.6)
    set.seed(1)
    connectParam <- list(list(mean=matrix(rbeta(4,.5,.5),2,2)),list(mean=matrix(rexp(4,.5),2,2)))
    model <- c("bernoulli","poisson")
    type <- "directed"
    mySampleMultiplexSBM_PB <-
       sampleMultiplexSBM(
       nbNodes = Nnodes,
        blockProp = blockProp,
       nbLayers = nbLayers,
       connectParam = connectParam,
       model=model,
       dimLabels = c('Individuals'), # generic name of the nodes
       type=type,
       seed = 1)
    #> [1] "use of sampleMultipartite"
    listSBM_PB <- mySampleMultiplexSBM_PB$listSBM
    names(listSBM_PB) <- c("Bernoulli","Poisson")
    plotMyMultiplexMatrix(listSBM_PB)

The resulting object is a list of SBM objects (of the SBM class). To build such an object from an observed matrix use the function `defineSBM`.

    listSBM_PB
    #> $Bernoulli
    #> Fit of a Simple Stochastic Block Model -- bernoulli variant
    #> =====================================================================
    #> Dimension = ( 40 ) - ( 0 ) blocks and no covariate(s).
    #> =====================================================================
    #> * Useful fields
    #>   $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
    #>   $blockProp, $connectParam, $covarParam, $covarList, $covarEffect
    #>   $expectation, $indMemberships, $memberships
    #> * R6 and S3 methods
    #>   $rNetwork, $rMemberships, $rEdges, plot, print, coef
    #> * Additional fields
    #>   $probMemberships, $loglik, $ICL, $storedModels,
    #> * Additional methods
    #>   predict, fitted, $setModel, $reorder
    #>
    #> $Poisson
    #> Fit of a Simple Stochastic Block Model -- poisson variant
    #> =====================================================================
    #> Dimension = ( 40 ) - ( 0 ) blocks and no covariate(s).
    #> =====================================================================
    #> * Useful fields
    #>   $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
    #>   $blockProp, $connectParam, $covarParam, $covarList, $covarEffect
    #>   $expectation, $indMemberships, $memberships
    #> * R6 and S3 methods
    #>   $rNetwork, $rMemberships, $rEdges, plot, print, coef
    #> * Additional fields
    #>   $probMemberships, $loglik, $ICL, $storedModels,
    #> * Additional methods
    #>   predict, fitted, $setModel, $reorder

- 2 Bipartite Gaussian dependent Layers and three clusters

    blockProp <- list(c(0.3, 0.3, 0.4), c(0.5, 0.5))
    Q <- sapply(blockProp, function(p) length(p))
    nbNodes <- c(80, 30)
    connectParam <- list()
    connectParam$mu <- vector("list", nbLayers)
    connectParam$mu[[1]] <- matrix(0.1, Q[1], Q[2]) + matrix(c(1, 1, 1, 0,
        1, 0), Q[1], Q[2])
    connectParam$mu[[2]] <- matrix(-2, Q[1], Q[2]) + matrix(c(1, 3, 2, 1, 2,
        3), Q[1], Q[2])
    connectParam$Sigma <- matrix(c(2, 1, 0.1, 4), nbLayers, nbLayers)
    model <- rep("gaussian", 2)
    mySampleMultiplexSBM_GG <-
      sampleMultiplexSBM(
         nbNodes = nbNodes,
         blockProp = blockProp,
         nbLayers = nbLayers,
         connectParam = connectParam,
         model = model,
         type = "bipartite",
         dependent = TRUE,
         dimLabels = c('row', 'col'),
         seed = 1)
    listSBM_GG <- mySampleMultiplexSBM_GG$listSBM
    plotMyMultiplexMatrix(listSBM_GG)

- 2 Bernoulli dependent Layers and 2 clusters

    ## MultiplexSBM Bernoulli with dependence
    Q <- 2
    set.seed(94)
    P00 <- matrix(runif(Q * Q), Q, Q)
    P10 <- matrix(runif(Q * Q), Q, Q)
    P01 <- matrix(runif(Q * Q), Q, Q)
    P11 <- matrix(runif(Q * Q), Q, Q)
    SumP <- P00 + P10 + P01 + P11
    P00 <- P00/SumP
    P01 <- P01/SumP
    P10 <- P10/SumP
    P11 <- P11/SumP
    connectParam <- list()
    connectParam$prob00 <- P00
    connectParam$prob01 <- P01
    connectParam$prob10 <- P10
    connectParam$prob11 <- P11
    model <- rep("bernoulli", 2)
    type <- "directed"
    nbLayers <- 2
    Nnodes <- 40
    blockProp <- c(0.6, 0.4)
    mySampleMultiplexSBM <-
       sampleMultiplexSBM(
         nbNodes = Nnodes,
         blockProp = blockProp,
         nbLayers = nbLayers,
         connectParam = connectParam,
         model = model,
         type = type,
         dependent = TRUE,
         seed = 1)
    listSBM_BB <- mySampleMultiplexSBM$listSBM
    plotMyMultiplexMatrix(listSBM_BB)

### Inference

We are now able to perform inference on the multiplex network (search of the “best” number of clusters in terms of ICL). To do so, we have to chose the model, i.e. the distribution used in each matrix and the dependence or independence between matrices conditionally to the clusters.

- 2 directed Layers : one Poisson, one Bernoulli and two clusters

    res_PB <- estimateMultiplexSBM(listSBM_PB)
    res_PB$storedModels

    #>   indexModel nbParams nbBlocks       ICL    loglik
    #> 1          1        9        2 -1926.202 -1894.847
    #> 2          2        2        1 -2362.585 -2355.207

One can now plot the reorganized matrices and the predicted values.

    plot(res_PB)

    plot(res_PB, type = 'expected')

One can also compare the estimated clusters to the simulated clusters. We recover the clusters perfectly.

    All <- plotAlluvial(list(simulated  = mySampleMultiplexSBM_PB$memberships$Individuals, estim = res_PB$memberships$Individuals))

    All
    #> $plotOptions
    #> $plotOptions$curvy
    #> [1] 0.3
    #>
    #> $plotOptions$alpha
    #> [1] 0.8
    #>
    #> $plotOptions$gap.width
    #> [1] 0.1
    #>
    #> $plotOptions$col
    #> [1] "darkolivegreen3"
    #>
    #> $plotOptions$border
    #> [1] "white"
    #>
    #>
    #> $tableFreq
    #>   simulated estim Freq
    #> 2         2     1   25
    #> 3         1     2   15

    -  2  Gaussian bipartite dependent Layers with 3 row clusters and 3 col clusters.

    res_GG <- estimateMultiplexSBM(listSBM_GG, dependent = TRUE, estimOptions = list(plot = FALSE, verbosity = 0 ))
    res_GG$storedModels

    #>   indexModel nbParams rowBlocks colBlocks nbBlocks       ICL    loglik
    #> 5          5       18         3         2        5 -9467.993 -9394.099
    #> 6          6       23         4         2        6 -9489.056 -9396.019
    #> 7          7       28         5         2        7 -9509.624 -9397.443
    #> 8          8       33         6         2        8 -9530.222 -9398.897
    #> 4          4       13         2         2        4 -9543.194 -9488.444
    #> 3          3        8         2         1        3 -9584.737 -9548.641
    #> 2          2        5         1         1        2 -9667.705 -9642.276
    #> 1          0       NA         0         0        0        NA        NA

    plot(res_GG)

- 2 Bernoulli dependent Layers

    res_BB <- estimateMultiplexSBM(listSBM_BB,dependent =  TRUE, estimOptions = list(plot = FALSE, verbosity = 0 ))
    res_BB$storedModels

    #>   indexModel nbParams nbBlocks       ICL    loglik
    #> 2          2       13        2 -1836.174 -1786.056
    #> 3          3       29        3 -1890.421 -1778.117
    #> 4          4       51        4 -1963.078 -1764.451
    #> 1          1        3        1 -2061.409 -2049.340

## References

Barbillon, Pierre, Sophie Donnet, Emmanuel Lazega, and Avner Bar-Hen. 2017. “Stochastic block models for multiplex networks: an application to a multilevel network of researchers.” Journal of the Royal Statistical Society Series A 180 (1): 295–314. https://ideas.repec.org/a/bla/jorssa/v180y2017i1p295-314.html.
Kéfi, Vincent AND Wieters, Sonia AND Miele. 2016. “How Structured Is the Entangled Bank? The Surprisingly Simple Organization of Multiplex Ecological Networks Leads to Increased Persistence and Resilience.” PLOS Biology 14 (8): 1–21. https://doi.org/10.1371/journal.pbio.1002527.