# Multipartite Stochastic Block Models

### An illustration on a mutualistic ecological network

#### team großBM

#### 2021-06-09

## Preliminaries

This vignette illustrates the use of the `estimateMultipartiteSBM` function and the methods accompanying the R6 classes `multipartiteSBMfit’.

### Requirements

The packages required for the analysis are sbm plus some others for data manipulation and representation:

    library(sbm)

## Dataset

We apply our methodology to an ecological mutualistic multipartite network.
The dataset –compiled and conducted by Dáttilo et al. (2016) at Centro de Investigaciones Costeras La Mancha (CICOLMA), located on the central coast of the Gulf of Mexico, Veracruz, Mexico– involves three general types of plant-animal mutualistic interaction: pollination, seed dispersal by frugivorous birds, and protective mutualisms between ants and plants with extrafloral nectaries.

The dataset –which is one of the largest compiled so far with respect to species richness, number of interactions and sampling effort– includes 4 functional groups, namely plants, pollinator species (referred as floral visitors), ant species and frugivorous bird species. Three binary bipartite networks have been collected representing interactions between 1/ plants and florals visitor, 2/ plants and ants, and 3/ plants and seed dispersal birds, resulting into three bipartite networks.

The FG are of respective sizes: \(n_1 = 141\) plant species, \(n_2 = 173\) pollinator species, \(n_3 = 46\) frugivorous bird species and \(n_4 = 30\) ant species.

The 3 networks contain \(753\) observed interactions of which \(55\%\) are plant-pollinator interactions, \(17\%\) are plant-birds interactions and \(28\%\) are plant-ant interactions.

    data(multipartiteEcologicalNetwork)
    str(multipartiteEcologicalNetwork)
    #> List of 3
    #>  $ Inc_plant_ant   : int [1:141, 1:30] 0 1 0 0 1 0 1 0 1 0 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : chr [1:141] "Acacia_cornigera" "Acacia_macracantha" "Achatocarpus_nigricans" "Agave_angustifolia" ...
    #>   .. ..$ : chr [1:30] "Camponotus_planatus" "Camponotus_mucronatus" "Paratrechina_longicornis_" "Crematogaster_brevispinosa" ...
    #>  $ Inc_plant_bird  : int [1:141, 1:46] 0 0 1 0 0 0 0 0 0 0 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : chr [1:141] "Acacia_cornigera" "Acacia_macracantha" "Achatocarpus_nigricans" "Agave_angustifolia" ...
    #>   .. ..$ : chr [1:46] "Cyanocorax_morio" "Tyrannus_forficatus" "Ortalis_vetula" "Myiozetetes_similis" ...
    #>  $ Inc_plant_flovis: int [1:141, 1:173] 0 0 0 0 0 0 0 1 1 0 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : chr [1:141] "Acacia_cornigera" "Acacia_macracantha" "Achatocarpus_nigricans" "Agave_angustifolia" ...
    #>   .. ..$ : chr [1:173] "Apis_melifera" "Lasioglossum_sp1" "Trigona_nigra" "Ascia_monuste" ...
    names(multipartiteEcologicalNetwork)
    #> [1] "Inc_plant_ant"    "Inc_plant_bird"   "Inc_plant_flovis"

### Formatting the data

We format the data to be able to use our functions i.e. we transform the matrices into an list containing the matrix, its type : `inc` for incidence matrix, `adj` for adjacency symmetric, and `diradj` for non symmetric (oriented) adjacency matrix, the name of functional group in row and the name of functional group in column. The three matrices are gathered in a list.

To do so, we use the function `defineNetwork`.

    Net <- multipartiteEcologicalNetwork
    type='bipartite'
    model = 'bernoulli'
    directed = FALSE
    PlantFlovis = defineSBM(Net$Inc_plant_flovis, model,type,directed,
                            dimLabels = c("Plants", "Flovis"))
    PlantAnt = defineSBM(Net$Inc_plant_ant,model,type,directed,
                          dimLabels = c("Plants", "Ants"))
    PlantBird = defineSBM(Net$Inc_plant_bird,model,type,directed,
                         dimLabels = c("Plants", "Birds"))

If one wants to keep a track of the names of the species, they should be used as rownames and colnames in the matrices.

    PlantFlovis$netMatrix[1:2,1:2]
    #> NULL

A plot of the data can be obtained with following command

    plotMyMultipartiteMatrix(list(PlantFlovis,PlantAnt,PlantBird))

### Mathematical Background

See Bar-Hen, P., and S. (To appear) for details.

### Inference

The model selection and the estimation are performed with the function `estimatemultipartiteBM`.

    estimOptions = list(initBM = FALSE)
    listSBM <- list(PlantFlovis, PlantAnt, PlantBird)
    myMSBM <- estimateMultipartiteSBM(listSBM, estimOptions)

contains the estimated parameters of the models we run through during the search of the better numbers of blocks.

RES_MBM contains the dataset and the results.

    myMSBM
    #> Fit of a Multipartite Stochastic Block Model
    #> 4 parts/functional groups ( Plants Flovis Ants Birds ),  3 networks
    #> =====================================================================
    #> nbNodes per FG = ( 141 173 30 46 ) --  nbBlocks per FG = ( 7 2 2 1 )
    #> distributions on each network:  bernoulli bernoulli bernoulli
    #> =====================================================================
    #> * Useful fields
    #>   $nbNetwork, $nbNodes, $nbBlocks, $dimLabels, $architecture
    #>   $modelName, $blockProp, $connectParam, $memberships, $networkData
    #>   $probMemberships, $loglik, $ICL, $storedModels,
    #> * R6 and S3 methods
    #>   plot, print, coef, predict, fitted, $setModel, $reorder

The best model has the following numbers of blocks

    myMSBM$nbBlocks
    #> Plants Flovis   Ants  Birds
    #>      7      2      2      1

To see the parameters estimated for the better model we use the following command `myMSBM$connectParam` or `myMSBM$blockProp`

    myMSBM$blockProp
    #> $Plants
    #> [1] 0.01418720 0.03802305 0.07840873 0.16063183 0.10614897 0.13507513 0.46752510
    #>
    #> $Flovis
    #> [1] 0.06221568 0.93778432
    #>
    #> $Ants
    #> [1] 0.1014381 0.8985619
    #>
    #> $Birds
    #> [1] 1
    myMSBM$connectParam
    #> [[1]]
    #> [[1]]$mean
    #>              [,1]         [,2]
    #> [1,] 2.464664e-15 4.440892e-16
    #> [2,] 1.001131e-15 4.440892e-16
    #> [3,] 1.651832e-01 3.427940e-02
    #> [4,] 4.173707e-03 8.566580e-08
    #> [5,] 1.917898e-01 6.378749e-02
    #> [6,] 8.826487e-07 3.317267e-04
    #> [7,] 9.573240e-02 7.490241e-03
    #>
    #>
    #> [[2]]
    #> [[2]]$mean
    #>              [,1]         [,2]
    #> [1,] 5.366301e-15 1.107167e-15
    #> [2,] 8.491907e-01 3.564830e-01
    #> [3,] 6.620266e-01 1.542027e-01
    #> [4,] 5.430946e-01 5.891885e-02
    #> [5,] 7.172257e-16 4.440892e-16
    #> [6,] 5.721732e-16 4.440892e-16
    #> [7,] 4.440892e-16 5.576222e-04
    #>
    #>
    #> [[3]]
    #> [[3]]$mean
    #>              [,1]
    #> [1,] 5.108074e-01
    #> [2,] 4.440892e-16
    #> [3,] 4.440892e-16
    #> [4,] 4.440892e-16
    #> [5,] 1.625224e-02
    #> [6,] 7.534085e-02
    #> [7,] 1.253527e-03

The clustering supplied by the better model are in `myMSBM$memberships***`.

    table(myMSBM$memberships$Plants)
    #>
    #>  1  2  3  4  5  6  7
    #>  2  5 11 23 15 20 65
    table(myMSBM$memberships$Ants)
    #>
    #>  1  2
    #>  3 27

    myMSBM$storedModels
    #>   indexModel nbParams nbBlocks Plants nbBlocks Flovis nbBlocks Ants
    #> 1          1       43               7               2             2
    #> 2          2       37               6               2             2
    #> 3          3       31               5               2             2
    #> 4          4       25               5               2             1
    #> 5          5       19               5               1             1
    #> 6          6       15               4               1             1
    #> 7          7       11               3               1             1
    #> 8          8        7               2               1             1
    #> 9          9        3               1               1             1
    #>   nbBlocks Birds nbBlocks       ICL    loglik
    #> 1              1       12 -2961.903 -2770.230
    #> 2              1       11 -2965.539 -2800.546
    #> 3              1       10 -2981.164 -2840.532
    #> 4              1        9 -3017.651 -2897.765
    #> 5              1        8 -3071.369 -2984.442
    #> 6              1        7 -3128.286 -3057.690
    #> 7              1        6 -3216.136 -3161.801
    #> 8              1        5 -3358.851 -3326.480
    #> 9              1        4 -3582.348 -3568.733

## Plots

We can either plot the reorganized matrix

    plot(myMSBM)

or the mesoscopic view

    plotOptions=list(vertex.size = c(12,6,4,4))
    plotOptions$vertex.shape = rep('circle',4)
    plotOptions$vertex.color = c('darkolivegreen3','darkgoldenrod2','salmon2','cadetblue2')
    plotOptions$edge.curved = 0.1
    plot(myMSBM,type = 'meso',plotOptions=plotOptions)

## References

Bar-Hen, A, Barbillon P., and Donnet S. To appear. “Block Models for Multipartite Networks.applications in Ecology and Ethnobiology.” Statistical Modelling, To appear.
Dáttilo, Wesley, Nubia Lara-Rodrı́guez, Pedro Jordano, Paulo R. Guimarães, John N. Thompson, Robert J. Marquis, Lucas P. Medeiros, Raul Ortiz-Pulido, Maria A. Marcos-Garcı́a, and Victor Rico-Gray. 2016. “Unravelling Darwin’s Entangled Bank: Architecture and Robustness of Mutualistic Networks with Multiple Interaction Types.” Proceedings of the Royal Society of London B: Biological Sciences 283 (1843).