Stochastic Block Models for Multiplex networks: principle

1. Preliminaries

This vignette illustrates the use of the estimateMultiplexSBM function and the methods accompanying the R6 class MultiplexSBM_fit.

The only package required for the analysis is sbm.

library(sbm)

2. 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.

3. Stochastic Block models for multiplex networks

3.1 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 $(Y^\ell_{ij})_{i,j = 1,\dots,n}$ , $\ell = 1, \dots, L$ .

The multiplex SBM

The $n$ nodes are divided into $Q$ blocks with proportions $\pi = (\pi_1, \dots, \pi_Q)$ , this clustering being encoded into the latent variables $(Z_i)_{i=1,\dots,n}$ such that $Z_i = q$ if node $i$ belongs to block $q$ . The relations between nodes $i$ and $j$ are driven by the blocks they belong to: $\begin{align*} (Z_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z_i = q) = \pi_q, \\ (Y^1_{ij}, \dots,Y^L_{ij})_{i \neq j} \text{ indep.} \mid (Z_i)_{i} \qquad & (Y^1_{ij}, \dots,Y^L_{ij}) \mid Z_i = q, Z_j = r \sim \mathcal{F}(\alpha_{qr}), \end{align*}$ where $\mathcal{F}$ is a $L$ dimensional probability distribution adapted to the data.

Examples

If all the networks are binary ( $Y^\ell_{ij} \in \{0,1\}$ ) then $\mathcal{F}$ is a $L$ -variate Bernoulli distribution and $\alpha_{qr} = (\alpha^{\omega}_{qr}, \omega \in \{0,1\}^L)$ with $\alpha^{\omega}_{qr} = \mathbb{P}\left((Y^1_{ij}, \dots,Y^L_{ij}) = \omega \mid Z_i = q, Z_j = r\right) \in [0,1]$ .
If all the networks are weighted with real values ( $Y^\ell_{ij} \in \mathbb{R}$ ) then $\mathcal{F}$ could be a $L$ -dimensional Gaussian distribution and $\alpha_{qr} = (\mu_{qr},\Sigma_{qr})$ with $\mu_{qr} \in \mathbb{R}^L$ , $\Sigma_{qr}$ a $L \times L$ 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.

3.2 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.

The multiplex SBM with independent layers

The clustering $(Z_i)_{i}$ is the same as above but, conditionally to the clusters, the layers are independent: for any $\ell,i,j$ , $Y^\ell_{ij} \mid Z_i = q, Z_j = r \sim_{ind} \mathcal{F}_\ell(\alpha^\ell_{qr}),$ where any $\mathcal{F}_\ell$ can be Bernoulli, Poisson, Gaussian or Zero Inflated Gaussian.

3.3 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.

The bipartite multiplex SBM

The $n_1$ row nodes are divided into $Q_1$ blocks with proportions $\pi^1$ and the $n_2$ column nodes into $Q_2$ blocks with proportions $\pi^2$ , encoded into the row and column clusterings $(Z^1_i)_{i=1,\dots,n_1}$ and $(Z^2_j)_{j=1,\dots,n_2}$ : $\begin{align*} (Z^1_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z^1_i = q) = \pi^1_q, \\ (Z^2_j)_{j} \text{ i.i.d.}, \text{ indep. of } (Z^1_i)_{i} \qquad & \mathbb{P}(Z^2_j = r) = \pi^2_r, \\ (Y^1_{ij}, \dots,Y^L_{ij})_{i,j} \text{ indep.} \mid (Z^1_i)_{i}, (Z^2_j)_{j} \qquad & (Y^1_{ij}, \dots,Y^L_{ij}) \mid Z^1_i = q, Z^2_j = r \sim \mathcal{F}(\alpha_{qr}), \end{align*}$ with the same distributions $\mathcal{F}$ as above (or independent layers $Y^\ell_{ij} \mid Z^1_i = q, Z^2_j = r \sim \mathcal{F}_\ell(\alpha^\ell_{qr})$ ).

3.4 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.

4. Data simulation

The function sampleMultiplexSBM 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 $2$ layers. The corresponding matrices can be plotted using the function plotMyMultiplexMatrix.

nbLayers <- 2

4.1 Two independent directed layers: one Bernoulli, one Poisson, 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 ) - ( 1 ) blocks.  No  edge covariate(s). No  node covariate(s). 
#> =====================================================================
#> * Useful fields 
#>   $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
#>   $blockProp, $connectParam, $covarParam, $covarList, $covarEffect 
#>   $nbNodesCovariates, $nodesCovariates, $nodesCovariatesParam 
#>   $expectation, $indMemberships, $memberships 
#> * R6 and S3 methods 
#>   $rNetwork, $rMemberships, $rEdges, plot(obj), print(obj), coef(obj) 
#> * Additional fields
#>   $probMemberships, $loglik, $ICL, $storedModels, 
#> * Additional methods 
#>   predict, fitted, $setModel, $reorder 
#> 
#> $Poisson
#> Fit of a Simple Stochastic Block Model -- poisson variant
#> =====================================================================
#> Dimension = ( 40 ) - ( 1 ) blocks.  No  edge covariate(s). No  node covariate(s). 
#> =====================================================================
#> * Useful fields 
#>   $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
#>   $blockProp, $connectParam, $covarParam, $covarList, $covarEffect 
#>   $nbNodesCovariates, $nodesCovariates, $nodesCovariatesParam 
#>   $expectation, $indMemberships, $memberships 
#> * R6 and S3 methods 
#>   $rNetwork, $rMemberships, $rEdges, plot(obj), print(obj), coef(obj) 
#> * Additional fields
#>   $probMemberships, $loglik, $ICL, $storedModels, 
#> * Additional methods 
#>   predict, fitted, $setModel, $reorder

4.2 Two dependent bipartite Gaussian layers: three row clusters, two column 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)

4.3 Two dependent directed Bernoulli layers: two 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)

5. 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.

5.1 Two independent directed layers: one Bernoulli, one Poisson

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')

or the graphon, i.e. the estimated connectivities alone, each block being represented by an interval whose length is its estimated proportion. The two layers following different distributions, the connectivities of each layer are divided by their maximum to be represented on the same color scale.

plot(res_PB, type = 'graphon')

One can also compare the estimated clusters to the simulated clusters:

table(mySampleMultiplexSBM_PB$memberships$Individuals, res_PB$memberships$Individuals)
#>    
#>      1  2
#>   1  0 15
#>   2 25  0

We recover the clusters perfectly, up to the label switching of the blocks.

5.2 Two dependent bipartite Gaussian layers

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 -9467.992 -9394.098
#> 6          6       23         4         2        6 -9487.963 -9394.926
#> 7          7       28         5         2        7 -9510.706 -9398.524
#> 8          8       33         6         2        8 -9534.096 -9402.771
#> 4          4       13         2         2        4 -9542.572 -9487.822
#> 3          3        8         2         1        3 -9584.782 -9548.686
#> 2          2        5         1         1        2 -9667.705 -9642.276
#> 1          0       NA         0         0        0        NA        NA

plot(res_GG)

plot(res_GG, type = 'graphon', plotOptions = list(legend = TRUE))

5.3 Two dependent directed Bernoulli 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.