Simple and Bipartite Stochastic Block Models • sbm

*Flammulina velutipes*, photo by Kork, license CC BY 3.0

1. Preliminaries

This vignette illustrates the use of the estimateSBM function and the methods accompanying the R6 classes SimpleSBMfit and BipartiteSBMfit.

1.1 Requirements

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

library(sbm)
library(ggplot2)
library(knitr)
library(patchwork) # to put figures side by side

To display the composition of the clusters, we use the following function, which gives a table with the names of the nodes of each cluster in a column:

clusterComposition <- function(memberships, nodeNames) {
  clusters <- split(nodeNames, factor(memberships, levels = 1:max(memberships)))
  maxSize  <- max(lengths(clusters))
  tab <- sapply(clusters, function(x) c(x, rep("", maxSize - length(x))))
  colnames(tab) <- paste("Cluster", seq_along(clusters))
  as.data.frame(tab)
}

1.2 Data set: antagonistic tree/fungus interaction network

We consider the fungus-tree interaction network studied by Vacher et al. (2008), available with the package sbm:

data("fungusTreeNetwork")
str(fungusTreeNetwork, max.level = 1)
#> List of 5
#>  $ tree_names  : Factor w/ 51 levels "Abies alba","Abies grandis",..: 1 2 3 14 42 4 5 6 7 8 ...
#>  $ fungus_names: Factor w/ 154 levels "Amphiporthe leiphaemia",..: 1 2 3 4 5 6 7 8 9 10 ...
#>  $ tree_tree   : num [1:51, 1:51] 0 12 9 3 2 2 0 0 2 7 ...
#>  $ fungus_tree : int [1:154, 1:51] 0 0 0 0 1 1 1 0 0 0 ...
#>  $ covar_tree  :List of 3

This data set provides information about 154 fungi sampled on 51 tree species. It is a list with the following entries:

fungus_names: list of the fungus species names
tree_names : list of the tree species names
fungus_tree : binary fungus-tree interactions
tree_tree : weighted tree-tree interactions (number of common fungal species two tree species host)
covar_tree : covariates associated to pairs of trees (namely genetic, taxonomic and geographic distances)

We first consider the tree-tree interactions resulting into a Simple Network. Then we consider the bipartite network between trees and fungi.

1.3 Mathematical Background

In all the models, each node $i$ belongs to a latent block $Z_i$ , and the edges are independent given the blocks, with a distribution depending only on the blocks of the nodes they involve. The model fitted in each section is recalled before the fit. The parameters are estimated with a variational EM algorithm, and the number of blocks is selected with the Integrated Classification Likelihood (ICL). See Leger (2016) and Jean-Benoist Leger (2015) for details.

2. Analysis of the tree/tree data

2.1 Tree-tree binary interaction networks

We first consider the binary network where an edge is drawn between two trees when they do share a least one common fungi:

Some names are long. For trees, the species of the group are given in parentheses (for instance “Cultivated Poplars (Populus trichocarpa, P. canescens, P.alba, P.nigra and their cultivated hybrids)”). For fungi, some names give the grouped species in parentheses (for instance “Botrytis cinerea (grouped with Botryotinia fuckeliana (anamorphic type))”) or a variety or special form (for instance “Coleosporium tussilaginis f.sp. senecionis-silvatici”). To keep the plots and tables readable, we only keep the part before the parentheses, and the genus and species for fungi: the short names still identify each tree and each fungus.

treeNames   <- trimws(sub("\\(.*", "", fungusTreeNetwork$tree_names))
fungusNames <- trimws(sub(" (var\\.|f\\.sp\\.) .*", "", sub("\\(.*", "", fungusTreeNetwork$fungus_names)))
tree_tree_binary <- 1 * (fungusTreeNetwork$tree_tree != 0)
colnames(tree_tree_binary) <- rownames(tree_tree_binary) <- treeNames

The simple function plotMyMatrix can be use to represent simple or bipartite SBM:

plotMyMatrix(tree_tree_binary, dimLabels =c('tree'))

We look for some latent structure of the network by adjusting a simple SBM with the function estimateSimpleSBM.

The Bernoulli SBM

The $n$ trees are divided into $Q$ blocks, with proportions $\pi = (\pi_1, \dots, \pi_Q)$ , and $Y_{ij} = 1$ if trees $i$ and $j$ share at least one fungus. The network being undirected, $Y_{ij} = Y_{ji}$ and the model is defined for $i < j$ : $\begin{align*} (Z_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z_i = q) = \pi_q, \\ (Y_{ij})_{i<j} \text{ indep.} \mid (Z_i)_{i} \qquad & Y_{ij} \mid Z_i = q, Z_j = l \sim \mathcal{B}ern(\alpha_{ql}), \end{align*}$ with $\alpha_{ql} = \alpha_{lq}$ the probability of connection between a tree of block $q$ and a tree of block $l$ .

mySimpleSBM <- tree_tree_binary %>% 
  estimateSimpleSBM("bernoulli", dimLabels = 'tree', estimOptions = list(verbosity = 0, plot = FALSE))

Once fitted, the user can manipulate the fitted model by accessing the various fields and methods enjoyed by the class simpleSBMfit. Most important fields and methods are recalled to the user via the show method:

class(mySimpleSBM)
#> [1] "SimpleSBM_fit" "SimpleSBM"     "SBM"           "R6"
mySimpleSBM
#> Fit of a Simple Stochastic Block Model -- bernoulli variant
#> =====================================================================
#> Dimension = ( 51 ) - ( 5 ) 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

For instance,

mySimpleSBM$nbBlocks
#> [1] 5
mySimpleSBM$nbNodes
#> tree 
#>   51
mySimpleSBM$nbCovariates
#> [1] 0

The plot method is available as a S3 or R6 method. The default represents the network data reordered according to the memberships estimated in the SBM. The red lines separate the blocks, and the background of each block is colored with an opacity growing with the mean value of the entries of the block: in the Bernoulli model, it is the estimated probability of connection $\widehat{\alpha}_{ql}$ between the blocks $q$ and $l$ .

The estimated connectivity can also be represented alone, as a graphon: each block is represented by an interval of $[0,1]$ whose length is its estimated proportion $\widehat{\pi}_q$ , and the square $q \times l$ is colored according to $\widehat{\alpha}_{ql}$ , with the same colors as the background of the data plot. We put the two plots side by side with the package patchwork.

plot(mySimpleSBM, type = "data", plotOptions = list(rowNames = TRUE, colNames = TRUE)) |
  plot(mySimpleSBM, type = "graphon")

The parameters of the graphon can be displayed as follows:

coef(mySimpleSBM, 'block')
#> [1] 0.3320954 0.1766383 0.1378665 0.1766999 0.1766999
coef(mySimpleSBM, 'connectivity')
#> $mean
#>            [,1]      [,2]       [,3]       [,4]       [,5]
#> [1,] 0.98230909 0.9828930 0.92749023 0.60851311 0.05994511
#> [2,] 0.98289303 0.9937459 0.29390960 0.91904879 0.22617761
#> [3,] 0.92749023 0.2939096 0.79890626 0.05802394 0.02169247
#> [4,] 0.60851311 0.9190488 0.05802394 0.47209785 0.06626655
#> [5,] 0.05994511 0.2261776 0.02169247 0.06626655 0.02899357

About model selection and choice of the number of blocks

During the estimation, a certain range of models are explored corresponding to different number of blocks. By default, the best model in terms of Integrated Classification Likelihood is sent back. In fact, all the model are stored internally. The user can have a quick glance at them via the $storedModels field:

mySimpleSBM$storedModels %>% kable()

indexModel	nbParams	nbBlocks	ICL	loglik
1	1	1	-883.3334	-879.7581
2	4	2	-619.0799	-606.3880
3	8	3	-537.5179	-512.1339
4	13	4	-540.6318	-498.9806
5	19	5	-520.6645	-459.1706
6	26	6	-530.6278	-445.7159
7	34	7	-544.0191	-432.1138
8	43	8	-562.1450	-419.6710

We can then see what models are competitive in terms of model selection by checking the ICL:

mySimpleSBM$storedModels %>%  ggplot() + aes(x = nbBlocks, y = ICL)  + geom_line() + geom_point(alpha = 0.5)

The 4-block or 6-block model could have been a “reasonable” choice too, in place of the 5-block model. The user can update the current simpleSBMfit thanks to the setModel.

We use it to compare the clusterings obtained with 4, 5 and 6 blocks with an alluvial plot (function plotAlluvial): each vertical axis represents a clustering, and the stripes show how the trees move from one cluster to another. The stripes are colored according to the clusters of the first clustering, which shows how the clusters split when the number of blocks increases.

clust_5blocks <-mySimpleSBM$memberships
mySimpleSBM$setModel(4)
clust_4blocks <-mySimpleSBM$memberships
mySimpleSBM$setModel(6)
clust_6blocks <-mySimpleSBM$memberships
mySimpleSBM$setModel(5)
plotAlluvial(list('4 blocks'=clust_4blocks,'5 blocks'=clust_5blocks,'6 blocks'=clust_6blocks))

We observe that going from 4 blocks to 5 blocks roughly divides block 1 into two clusters. Similarly, going from 5 blocks to 6 blocks, roughly divides block 4 into two smaller blocks.

2.2 Analysis of the weighted interaction network

Instead of considering the binary network tree-tree we may consider the weighted network where the link between two trees is the number of fungi they share.

We plot the matrix with function plotMyMatrix:

tree_tree <- fungusTreeNetwork$tree_tree
colnames(tree_tree) <- rownames(tree_tree) <- treeNames
plotMyMatrix(tree_tree, dimLabels = c('tree'))

Here again, we look for some latent structure of the network by adjusting a simple SBM with the function estimateSimpleSBM, considering a Poisson distribution on the edges.

The Poisson SBM

$Y_{ij}$ is now the number of fungi shared by trees $i$ and $j$ . The blocks are defined as before, and $\begin{align*} (Z_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z_i = q) = \pi_q, \\ (Y_{ij})_{i<j} \text{ indep.} \mid (Z_i)_{i} \qquad & Y_{ij} \mid Z_i = q, Z_j = l \sim \mathcal{P}(\lambda_{ql}), \end{align*}$ with $\lambda_{ql} = \lambda_{lq}$ the mean number of fungi shared by a tree of block $q$ and a tree of block $l$ .

mySimpleSBMPoisson <- tree_tree %>% 
  estimateSimpleSBM("poisson", dimLabels = 'tree', estimOptions = list(verbosity = 0, plot = FALSE))

class(mySimpleSBMPoisson)
#> [1] "SimpleSBM_fit" "SimpleSBM"     "SBM"           "R6"
mySimpleSBMPoisson
#> Fit of a Simple Stochastic Block Model -- poisson variant
#> =====================================================================
#> Dimension = ( 51 ) - ( 6 ) 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

For instance,

mySimpleSBMPoisson$nbBlocks
#> [1] 6
mySimpleSBMPoisson$nbNodes
#> tree 
#>   51
mySimpleSBMPoisson$nbCovariates
#> [1] 0

We now plot the matrix reordered according to the memberships estimated in the SBM, next to the corresponding graphon. The entries are now counts, represented in gray levels, and the background of each block corresponds to the estimated mean number of shared fungi $\widehat{\lambda}_{ql}$ between the blocks $q$ and $l$ .

plot(mySimpleSBMPoisson, type = "data", plotOptions = list(rowNames = TRUE, colNames = TRUE)) |
  plot(mySimpleSBMPoisson, type = "graphon")

The graphon can also be drawn as a network, in a mesoscopic view: each block is a node whose area is proportional to the block proportion $\widehat{\pi}_q$ , and the mean number of shared fungi $\widehat{\lambda}_{ql}$ between two blocks is represented by an edge whose width and color grow with $\widehat{\lambda}_{ql}$ , with the same colors as the graphon. The connectivity within a block, $\widehat{\lambda}_{qq}$ , is represented by a loop. The layout being random, it is fixed with the option seed.

plot(mySimpleSBMPoisson, type = "meso", plotOptions = list(seed = 1))

The composition of the clusters can now be scrutinized:

kable(clusterComposition(mySimpleSBMPoisson$memberships, treeNames),
      caption = "Tree clusters (Poisson SBM)")

Tree clusters (Poisson SBM)
Cluster 1	Cluster 2	Cluster 3	Cluster 4	Cluster 5	Cluster 6
Abies alba	Large Maples	Abies nordmanniana	Small Maples	Cupressus sempervirens	Betulus spp
Abies grandis	Fagus silvatica	Larix kaempferi	Alnus glutinosa	Pinus brutia	Carpinus betulus
Cedrus spp	Fraxinus spp	Picea sitchensis	Castanea sativa	Pinus cembra	Populus tremula
Larix decidua	Juglans spp	Pinus halepensis	Quercus ilex	Pinus pinea	Quercus pyrenaica
Picea excelsa	Cultivated Poplars	Pinus nigra nigra	Quercus pubescens	Pinus radiata	Sorbus aucuparia
Pinus nigra laricio	Prunus avium	Pinus strobus	Quercus suber	Pinus taeda	Sorbus domestica
Pinus pinaster	Quercus petraea	Pinus uncinata	Sorbus aria	Platanus hybrida	Taxus baccata
Pinus sylvestris	Quercus robur		Tilia spp	Tsuga heterophylla	Thuja plicata
Pseudotsuga menziesii	Quercus rubra				Ulmus spp
	Sorbus torminalis

Comparison of the binary and weighted clusterings

We compare the clusterings obtained on the binary network (Bernoulli SBM) and on the weighted network (Poisson SBM) with an alluvial plot, the stripes being colored according to the clusters of the Bernoulli SBM.

plotAlluvial(list(Bernoulli = mySimpleSBM$memberships,
                  Poisson   = mySimpleSBMPoisson$memberships))

The most connected cluster of the Bernoulli SBM is cluster 1: its 17 trees share at least one fungus with 37 other trees on average (out of 50), and the probability of connection within the cluster is $\widehat{\alpha}_{1 1} =$ 0.98. In the binary network, these trees are almost all connected to each other, so the network cannot distinguish them any further. The weighted network, which counts the shared fungi, reveals a finer structure: this cluster is split by the Poisson SBM into

9 trees (Poisson cluster 1) sharing on average $\widehat{\lambda} =$ 9 fungi with each other: Abies alba, Abies grandis, Cedrus spp, Larix decidua, Picea excelsa, Pinus nigra laricio, Pinus pinaster, Pinus sylvestris, Pseudotsuga menziesii;
5 trees (Poisson cluster 3) sharing fewer fungi with each other ( $\widehat{\lambda} =$ 3.9): Abies nordmanniana, Larix kaempferi, Picea sitchensis, Pinus nigra nigra, Pinus uncinata;
and 3 trees moving to other clusters: Juglans spp, Sorbus aria, Pinus brutia.

Both main parts are made of conifers: the binary network gathers the conifers, and the weighted network separates those sharing many fungi from those sharing fewer.

2.3 Introduction of covariates

We have on each pair of trees 3 covariates, namely the genetic distance, the taxonomic distance and the geographic distance. Each covariate has to be introduced as a matrix: $X^k_{ij}$ corresponds to the value of the $k$ -th covariate describing the couple $(i,j)$ .

The Poisson SBM with covariates on the edges

Let $x_{ij} = (X^1_{ij}, \dots, X^p_{ij})^\top \in \mathbb{R}^p$ be the vector of the $p = 3$ covariates of the pair $(i,j)$ . The covariates act on the mean of the edges, through a regression parameter $\beta \in \mathbb{R}^p$ common to all the pairs: $\begin{align*} (Z_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z_i = q) = \pi_q, \\ (Y_{ij})_{i<j} \text{ indep.} \mid (Z_i)_{i} \qquad & Y_{ij} \mid Z_i = q, Z_j = l \sim \mathcal{P}\left(\lambda_{ql} \exp(x_{ij}^\top \beta)\right). \end{align*}$ The blocks now capture the structure of the network which is not explained by the covariates.

mySimpleSBMCov<- 
  tree_tree %>% 
  estimateSimpleSBM(
    model = 'poisson', 
    directed = FALSE, 
    dimLabels = 'tree',
    covariates  = fungusTreeNetwork$covar_tree, 
    estimOptions = list(verbosity = 0, plot = FALSE, nbCores = 2)
  )

We select the best number of clusters (with respect to the ICL criteria)

mySimpleSBMCov$nbBlocks
#> [1] 4

We can now extract the parameters of interest, namely ( $\lambda$ , $\pi$ ) and the clustering of the nodes.

mySimpleSBMCov$connectParam
#> $mean
#>           [,1]       [,2]      [,3]      [,4]
#> [1,] 28.207064 14.8964888 5.9617522 1.2396157
#> [2,] 14.896489  9.5664171 3.7941904 0.4110826
#> [3,]  5.961752  3.7941904 1.6183797 0.3842726
#> [4,]  1.239616  0.4110826 0.3842726 0.1893782
mySimpleSBMCov$blockProp
#> [1] 0.3715916 0.2159544 0.2354116 0.1770425
mySimpleSBMCov$memberships
#>  [1] 1 1 2 1 2 2 4 4 2 1 3 1 1 1 1 2 1 2 3 3 2 1 2 1 3 3 2 1 3 2 3 1 4 1 1 3 1 3
#> [39] 4 1 1 3 2 4 4 1 4 4 3 3 4
mySimpleSBMCov$covarParam
#>            [,1]
#> [1,]  0.1975175
#> [2,] -2.0552576
#> [3,] -0.3586006

coef(mySimpleSBMCov, 'covariates')
#>            [,1]
#> [1,]  0.1975175
#> [2,] -2.0552576
#> [3,] -0.3586006

S3 methods are also available for fit and prediction (results hidden here)

#fitted(mySimpleSBMCov)
#predict(mySimpleSBMCov)
#predict(mySimpleSBMCov, fungusTreeNetwork$covar_tree)

3. Analysis of the tree/fungi data

We now analyze the bipartite tree/fungi interactions. The incidence matrix fungus_tree has the fungi in rows and the trees in columns. To get a wide plot, we put the trees in rows and the fungi in columns by transposing it: the bipartite SBM treats rows and columns in the same way, so this does not change the model. The incidence matrix can be plotted with the function plotMyMatrix:

tree_fungus <- t(fungusTreeNetwork$fungus_tree)
rownames(tree_fungus) <- treeNames
colnames(tree_fungus) <- fungusNames
plotMyMatrix(tree_fungus, dimLabels = c(row = 'tree', col = 'fungis'))

The bipartite SBM

The $n_1$ trees (in rows) are divided into $Q_1$ blocks with proportions $\pi^1$ , and the $n_2$ fungi (in columns) into $Q_2$ blocks with proportions $\pi^2$ . $Y_{ij} = 1$ if fungus $j$ was observed on tree $i$ , and $\begin{align*} (Z^1_i)_{i} \text{ i.i.d.} \qquad & \mathbb{P}(Z^1_i = q) = \pi^1_q, \\ (Z^2_j)_{j} \text{ i.i.d.}, \text{ indep. of } (Z^1_i)_{i} \qquad & \mathbb{P}(Z^2_j = r) = \pi^2_r, \\ (Y_{ij})_{i,j} \text{ indep.} \mid (Z^1_i)_{i}, (Z^2_j)_{j} \qquad & Y_{ij} \mid Z^1_i = q, Z^2_j = r \sim \mathcal{B}ern(\alpha_{qr}), \end{align*}$ with $\alpha_{qr}$ the probability that a fungus of block $r$ is observed on a tree of block $q$ .

The model is fitted with the function estimateBipartiteSBM:

myBipartiteSBM <- 
  tree_fungus %>% 
  estimateBipartiteSBM(model = 'bernoulli', dimLabels = c('tree', 'fungis'), estimOptions = list(verbosity = 0, plot = FALSE))

myBipartiteSBM$nbNodes
#>   tree fungis 
#>     51    154
myBipartiteSBM$nbBlocks
#>   tree fungis 
#>      4      4
myBipartiteSBM$connectParam
#> $mean
#>            [,1]      [,2]        [,3]        [,4]
#> [1,] 0.96798282 0.5281051 0.327457450 0.020134849
#> [2,] 0.07714501 0.5896776 0.003883283 0.155064928
#> [3,] 0.84173863 0.2393528 0.099918358 0.001333581
#> [4,] 0.06797475 0.1119710 0.005829596 0.019186990
coef(myBipartiteSBM, 'block')
#> $tree
#> [1] 0.1034934 0.1956584 0.2489919 0.4518563
#> 
#> $fungis
#> [1] 0.02655516 0.05376727 0.31487519 0.60480238
coef(myBipartiteSBM, 'connectivity')
#> $mean
#>            [,1]      [,2]        [,3]        [,4]
#> [1,] 0.96798282 0.5281051 0.327457450 0.020134849
#> [2,] 0.07714501 0.5896776 0.003883283 0.155064928
#> [3,] 0.84173863 0.2393528 0.099918358 0.001333581
#> [4,] 0.06797475 0.1119710 0.005829596 0.019186990

We can now plot the reorganized matrix (top). The row blocks (trees) and the column blocks (fungi) are separated by the red lines, the background of each block representing the estimated probability of connection $\widehat{\alpha}_{qr}$ between the tree block $q$ and the fungus block $r$ . As in the simple case, the estimated connectivity can be represented as a graphon (bottom), the row and column blocks being sized by their estimated proportions. The plots being wide, we put them one above the other with patchwork.

plot(myBipartiteSBM, type = "data") / plot(myBipartiteSBM, type = "graphon")

In the mesoscopic view of a bipartite SBM, the blocks of trees (on top) and the blocks of fungi (at the bottom) are drawn with different colors, and each edge represents the probability of connection $\widehat{\alpha}_{qr}$ between a block of trees and a block of fungi:

plot(myBipartiteSBM, type = "meso")

The clusters of trees are:

kable(clusterComposition(myBipartiteSBM$memberships$tree, rownames(tree_fungus)),
      caption = "Tree clusters (bipartite SBM)")

Tree clusters (bipartite SBM)
Cluster 1	Cluster 2	Cluster 3	Cluster 4
Picea excelsa	Large Maples	Abies alba	Small Maples
Pinus nigra laricio	Castanea sativa	Abies grandis	Alnus glutinosa
Pinus pinaster	Fagus silvatica	Abies nordmanniana	Betulus spp
Pinus sylvestris	Fraxinus spp	Cedrus spp	Carpinus betulus
Pseudotsuga menziesii	Juglans spp	Larix decidua	Cupressus sempervirens
	Cultivated Poplars	Larix kaempferi	Pinus brutia
	Prunus avium	Picea sitchensis	Pinus cembra
	Quercus petraea	Pinus halepensis	Pinus radiata
	Quercus robur	Pinus nigra nigra	Platanus hybrida
	Quercus rubra	Pinus pinea	Populus tremula
		Pinus strobus	Quercus ilex
		Pinus taeda	Quercus pubescens
		Pinus uncinata	Quercus pyrenaica
			Quercus suber
			Sorbus aria
			Sorbus aucuparia
			Sorbus domestica
			Sorbus torminalis
			Taxus baccata
			Thuja plicata
			Tilia spp
			Tsuga heterophylla
			Ulmus spp

and the clusters of fungi are:

kable(clusterComposition(myBipartiteSBM$memberships$fungis, colnames(tree_fungus)),
      caption = "Fungus clusters (bipartite SBM)")

Fungus clusters (bipartite SBM)
Cluster 1	Cluster 2	Cluster 3	Cluster 4
Armillaria ostoyae	Armillaria gallica	Armillaria cepistipes	Amphiporthe leiphaemia
Heterobasidion annosum	Armillaria mellea	Caliciopsis pinea	Apiognomonia errabunda
Sphaeropsis sapinea	Botrytis cinerea	Cenangium ferruginosum	Apiognomonia veneta
Sydowia polyspora	Botryosphaeria stevensii	Ceratocystis ips	Asteroma tiliae
	Fomitopsis pinicola	Ceratocystis minor	Biscogniauxia mediterranea
	Haematonectria haematococca	Chrysomyxa abietis	Bjerkandera adusta
	Ophiostoma piceae	Chrysomyxa ledi	Blumeriella jaapii
	Phaeolus schweinitzii	Coleosporium tussilaginis	Botryosphaeria dothidea
		Cronartium flaccidum	Ceratocystis platani
		Cronartium ribicola	Chondrostereum purpureum
		Crumenulopsis sororia	Colpoma quercinum
		Cyclaneusma minus	Cristulariella depraedans
		Cyclaneusma niveum	Cronartium quercuum
		Cylindrocarpon didymum	Cryptodiaporthe castanea
		Delphinella abietis	Cryptostroma corticale
		Gremmeniella abietina	Cryptodiaporthe hystrix
		Herpotrichia juniperi	Cryphonectria parasitica
		Lachnellula willkommii	Cryptodiaporthe populea
		Leptographium lundbergii	Daedaleopsis confragosa
		Leptographium procerum	Dicarpella dryina
		Lirula macrospora	Discella castanea
		Lirula nervisequia	Drepanopeziza punctiformis
		Lophodermium conigenum	Entoleuca mammata
		Lophodermium piceae	Eutypa lata
		Lophodermium pinastri	Fistulina hepatica
		Lophodermium seditiosum	Fomes fomentarius
		Melampsorella caryophyllacearum	Fomitopsis cytisina
		Melampsora populnea	Fusarium oxysporum
		Mycosphaerella pini	Fusicoccum quercus
		Nectria fuckeliana	Ganoderma adspersum
		Ophiostoma serpens	Ganoderma applanatum
		Pestalotiopsis funerea	Ganoderma resinaceum
		Phacidium coniferarum	Gnomonia leptostyla
		Phaeocryptopus gaeumannii	Gymnopus fusipes
		Phellinus chrysoloma	Hypoxylon fragiforme
		Phellinus hartigii	Laetiporus sulphureus
		Phellinus pini	Leucostoma persoonii
		Pucciniastrum epilobii	Melampsora allii-populina
		Rhabdocline pseudotsugae	Melampsora laricis-populina
		Rhizosphaera kalkhoffii	Melanconis modonia
		Rhizosphaera macrospora	Meripilus giganteus
		Rhizosphaera oudemansii	Microsphaera alphitoides
		Rhizina undulata	Microstroma juglandis
		Sarea resinae	Monochaetia monochaeta
		Sirococcus strobilinus	Mycosphaerella castaneicola
		Sparassis crispa	Mycosphaerella dearnessii
		Stereum sanguinolentum	Mycosphaerella microsora
		Truncatella hartigii	Nectria cinnabarina
		Valsa kunzei	Nectria coccinea
		Zythiostroma pinastri	Nectria ditissima
			Nectria veuillotiana
			Neonectria galligena
			Ophiostoma ulmi
			Oudemansiella mucida
			Oxyporus populinus
			Pezicula cinnamomea
			Phellinus igniarius
			Phellinus robustus
			Phloeospora aceris
			Phomopsis juniperivora
			Phomopsis quercella
			Phyllosticta aceris
			Phyllactinia guttata
			Pleurotus ostreatus
			Pleuroceras pseudoplatani
			Podosphaera clandestina
			Pollaccia radiosa
			Polyporus squamosus
			Pseudoinonotus dryadeus
			Rhytisma acerinum
			Rosellinia necatrix
			Sawadaea tulasnei
			Schizophyllum commune
			Seiridium cardinale
			Septotis podophyllina
			Septoria quercicola
			Stegonsporium pyriforme
			Stereum hirsutum
			Stigmina carpophila
			Taphrina betulina
			Taphrina caerulescens
			Taphrina cerasi
			Taphrina populina
			Titaeosporina tremulae
			Trametes gibbosa
			Trametes versicolor
			Valsa ambiens
			Valsa nivea
			Valsa sordida
			Venturia fraxini
			Venturia inaequalis
			Verticillium dahliae

References

Jean-Benoist Leger, Corinne Vacher, Jean-Jacques Daudin. 2015. “Clustering Methods Differ in Their Ability to Detect Patterns in Ecological Networks.” Methods in Ecology and Evolution.

Leger, Jean-Benoist. 2016. “Blockmodels: A r-Package for Estimating in Latent Block Model and Stochastic Block Model, with Various Probability Functions, with or Without Covariates.” arXiv Preprint arXiv:1602.07587.

Vacher, Corinne, Dominique Piou, and Marie-Laure Desprez-Loustau. 2008. “Architecture of an Antagonistic Tree/Fungus Network: The Asymmetric Influence of Past Evolutionary History.” PloS One 3 (3): e1740.