Data

$n = 141$ plant species,
$p^1 = 173$ floral visitor species (insects),
$p^2 = 30$ ant species,
$p^3 = 46$ seed dispersor species (birds).

For $1 \leq h \leq 3$ categories of animals, \[ Y^h = [Y^h_{ij}]_{1 \leq i \leq n, 1 \leq j \leq p^h} \] \[ Y^h_{ij} = \mathbb{I}\left\{\begin{array}{c} \text{an interaction between plant species $i$ and} \\ \text{animal species $j$ from category $h$ has been observed} \end{array} \right\} \]

Multipartite LBM

Model

\[\begin{align} (Z_i)_{1 \leq i \leq n} & \text{ iid}: & Z_i & \sim \mathcal{M}_K(1, \pi) \nonumber \\ \text{for } 1 \leq h \leq H, \quad (W^h_j)_{1 \leq j \leq p^h} & \text{ iid}: & W^h_j & \sim \mathcal{M}_{L_h}(1, \rho^h) \\ (Y^h_{ij})_{1 \leq i \leq n, 1 \leq j \leq p, 1 \leq h \leq H} & \text{ indep.} \mid (Z_i), (W^h_j): & (Y^h_{ij} \mid Z_i=k, W^h_j=\ell) & \sim \mathcal{B}(\gamma^h_{k\ell}) \nonumber \end{align}\]

Model selection

##       Plants Flovis Ants Birds       ICL
##  [1,]      7      2    2     1 -2961.903
##  [2,]      6      2    2     1 -2965.539
##  [3,]      5      2    2     1 -2981.164
##  [4,]      5      2    1     1 -3017.651
##  [5,]      5      1    1     1 -3071.369
##  [6,]      4      1    1     1 -3128.286
##  [7,]      3      1    1     1 -3216.136
##  [8,]      2      1    1     1 -3358.851
##  [9,]      1      1    1     1 -3582.348

Parameters

## [1] "number of groups"

## Plants Flovis   Ants  Birds 
##      7      2      2      1

## [1] "group proportions for plants"

## [1] 0.46752510 0.16063183 0.13507513 0.07840873 0.10614897 0.01418720 0.03802305

## [1] "group proportions for insects"

## [1] 0.06221568 0.93778432

## [1] "plant - insect connectivity matrix"

##              [,1]         [,2]
## [1,] 9.573240e-02 7.490241e-03
## [2,] 4.173707e-03 8.566580e-08
## [3,] 8.826487e-07 3.317267e-04
## [4,] 1.651832e-01 3.427940e-02
## [5,] 1.917898e-01 6.378749e-02
## [6,] 2.464664e-15 4.440892e-16
## [7,] 1.001131e-15 4.440892e-16

Modelling a multipartite plant-animals network with LBMs

S. Robin

feb 2022