Multipartite Stochastic Block Models
In sbm: Stochastic Blockmodels

knitr::opts_chunk$set(
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Preliminaries

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

Requirements

The only package required for the analysis is sbm:

library(sbm)

Dataset

We apply our methodology to an ecological mutualistic multipartite network.
The dataset --compiled and conducted by @Dattilo 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 (FG), 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)
names(multipartiteEcologicalNetwork)

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]

A plot of the data can be obtained with following command

plotMyMultipartiteMatrix(list(PlantFlovis, PlantAnt, PlantBird))

Mathematical Background

See @multipartite for details.

Inference

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

load('resMultipartiteEcological.rda')

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

\code{myMSBM} contains the estimated parameters of the models we run through during the search of the better numbers of blocks.

myMSBM

The best model has the following numbers of blocks:

myMSBM$nbBlocks

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

myMSBM$blockProp
myMSBM$connectParam

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

table(myMSBM$memberships$Plants)
table(myMSBM$memberships$Ants)

myMSBM$storedModels

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)