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This vignette illustrates the use of the estimateSBM
function and the methods accompanying the R6 classes
SimpleSBMfit
and BipartiteSBMfit
.
The packages required for the analysis are sbm plus some others for data manipulation and representation:
We consider the fungus-tree interaction network studied by Vacher, Piou, and Desprez-Loustau (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:
fungi_list
: list of the fungus species namestree_list
: list of the tree species namesfungus_tree
: binary fungus-tree interactionstree_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.
See Leger (2016) for details.
We first consider the binary network where an edge is drawn between two trees when they do share a least one common fungi:
The simple function plotMyMatrix
can be use to represent
simple or bipartite SBM:
We look for some latent structure of the network by adjusting a
simple SBM with the function estimateSimpleSBM
.
mySimpleSBM <- tree_tree_binary %>%
estimateSimpleSBM("bernoulli", dimLabels = 'tree', estimOptions = list(verbosity = 1, plot = TRUE))
#> -> Estimation for 1 groups
#>
#> -> Computation of eigen decomposition used for initalizations
#>
#> -> Pass 1
#>
#> -> Pass 2
#>
#> -> Pass 3
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 and no covariate(s).
#> =====================================================================
#> * Useful fields
#> $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
#> $blockProp, $connectParam, $covarParam, $covarList, $covarEffect
#> $expectation, $indMemberships, $memberships
#> * R6 and S3 methods
#> $rNetwork, $rMemberships, $rEdges, plot, print, coef
#> * Additional fields
#> $probMemberships, $loglik, $ICL, $storedModels,
#> * Additional methods
#> predict, fitted, $setModel, $reorder
For instance,
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.
One can also plot the expected network which, in case of the Bernoulli model, corresponds to the probability of connection between any pair of nodes in the network.
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
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 $storedModels
field:
indexModel | nbParams | nbBlocks | ICL | loglik |
---|---|---|---|---|
1 | 1 | 1 | -883.3334 | -879.7581 |
2 | 4 | 2 | -619.0799 | -606.3880 |
3 | 8 | 3 | -537.5179 | -512.1339 |
4 | 13 | 4 | -540.6318 | -498.9806 |
5 | 19 | 5 | -520.6645 | -459.1706 |
6 | 26 | 6 | -530.6278 | -445.7159 |
7 | 34 | 7 | -544.0191 | -432.1138 |
8 | 43 | 8 | -562.1450 | -419.6710 |
We can then see what models are competitive in terms of model selection by checking the ICL:
mySimpleSBM$storedModels %>% ggplot() + aes(x = nbBlocks, y = ICL) + geom_line() + geom_point(alpha = 0.5)
The 4-block model could have been a good choice too, in place of the
5-block model. The user can update the current simpleSBMfit
thanks to the the setModel
method:
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
:
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.
mySimpleSBMPoisson <- tree_tree %>%
estimateSimpleSBM("poisson", 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(mySimpleSBMPoisson)
#> [1] "SimpleSBM_fit" "SimpleSBM" "SBM" "R6"
mySimpleSBMPoisson
#> Fit of a Simple Stochastic Block Model -- poisson variant
#> =====================================================================
#> Dimension = ( 51 ) - ( 6 ) blocks and no covariate(s).
#> =====================================================================
#> * Useful fields
#> $nbNodes, $modelName, $dimLabels, $nbBlocks, $nbCovariates, $nbDyads
#> $blockProp, $connectParam, $covarParam, $covarList, $covarEffect
#> $expectation, $indMemberships, $memberships
#> * R6 and S3 methods
#> $rNetwork, $rMemberships, $rEdges, plot, print, coef
#> * 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:
One can also plot the expected network which, in case of the Poisson model, corresponds to the expectation of connection between any pair of nodes in the network.
The same manipulations can be made on the models as before.
coef(mySimpleSBMPoisson, 'block')
#> [1] 0.1763561 0.1957351 0.1375982 0.1569772 0.1569772 0.1763561
coef(mySimpleSBMPoisson, 'connectivity')
#> $mean
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 9.0286270 3.2910355 6.04727443 1.06850270 1.85464212 0.11062572
#> [2,] 3.2910355 5.9646020 1.33637565 2.17610068 0.49999123 0.26547703
#> [3,] 6.0472744 1.3363757 3.94065914 0.40263458 1.46245649 0.07080814
#> [4,] 1.0685027 2.1761007 0.40263458 0.82166374 0.14089332 0.07793391
#> [5,] 1.8546421 0.4999912 1.46245649 0.14089332 0.54121406 0.02263218
#> [6,] 0.1106257 0.2654770 0.07080814 0.07793391 0.02263218 0.02940543
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 \((i,j)\).
mySimpleSBMCov<-
tree_tree %>%
estimateSimpleSBM(
model = 'poisson',
directed = FALSE,
dimLabels = 'tree',
covariates = fungusTreeNetwork$covar_tree,
estimOptions = list(verbosity = 0, plot = FALSE, nbCores = 2)
)
mySimpleSBMCov$connectParam
#> $mean
#> [,1] [,2] [,3] [,4]
#> [1,] 28.194994 14.8895579 5.9602485 1.2392858
#> [2,] 14.889558 9.5641591 3.7929799 0.4101411
#> [3,] 5.960249 3.7929799 1.6180940 0.3841918
#> [4,] 1.239286 0.4101411 0.3841918 0.1892972
mySimpleSBMCov$blockProp
#> [1] 0.3715916 0.2159544 0.2354118 0.1770423
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,] 0.1975890
#> [2,] -2.0549756
#> [3,] -0.3583139
S3 methods are also available for fit and prediction (results hidden here)
We now analyze the bipartite tree/fungi interactions. The incidence matrix can be plotted with the function
myBipartiteSBM <-
fungusTreeNetwork$fungus_tree %>%
estimateBipartiteSBM(model = 'bernoulli', dimLabels = c('fungis', 'tree'),estimOptions = list(verbosity = 0, plot = FALSE))
myBipartiteSBM$nbNodes
#> fungis tree
#> 154 51
myBipartiteSBM$nbBlocks
#> fungis tree
#> 4 4
myBipartiteSBM$connectParam
#> $mean
#> [,1] [,2] [,3] [,4]
#> [1,] 0.96813478 0.077538579 0.840370657 0.067563355
#> [2,] 0.52055882 0.584398216 0.230893917 0.107930384
#> [3,] 0.32450427 0.003624764 0.098526840 0.005780612
#> [4,] 0.01834547 0.154334411 0.001330278 0.019219920
coef(myBipartiteSBM, 'block')
#> $row
#> [1] 0.02655516 0.05570334 0.31666508 0.60107642
#>
#> $col
#> [1] 0.1050494 0.1963968 0.2477304 0.4508234
coef(myBipartiteSBM, 'connectivity')
#> $mean
#> [,1] [,2] [,3] [,4]
#> [1,] 0.96813478 0.077538579 0.840370657 0.067563355
#> [2,] 0.52055882 0.584398216 0.230893917 0.107930384
#> [3,] 0.32450427 0.003624764 0.098526840 0.005780612
#> [4,] 0.01834547 0.154334411 0.001330278 0.019219920
We can now plot the reorganized matrix.
plot(myBipartiteSBM, type = 'meso',
dimLabels = list(row = 'fungis', col= 'tree'),
plotOptions = list(edge.width = 1, vertex.size = c(1,2)))
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.