We present the performances of GREMLINS on a simulated multipartite network. GREMLINS includes a function rMBM to simulate multipartite networks. Mathematical details can be found in @multipartite.
We use the function rMBM provided in the package to simulate a multipartite network involving \(2\) functional groups (namely A and B) of respective sizes
\[n_A = 60, \quad, n_B = 50.\]
A and B are divided respectively into \(3\) and \(2\) blocks. The sizes of the blocks are generated randomly. For reproductibility, we fix the random seed to an arbitrarily chosen value.
namesFG <- c('A','B')
v_NQ <- c(60,50) #size of each FG
list_pi = list(c(0.16 ,0.40 ,0.44),c(0.3,0.7)) #proportion of each block in each FG
list_pi[[1]]
#> [1] 0.16 0.40 0.44
We assume that we observe \(3\) interactions matrices
- A-B : continuous weighted interactions
- B-B : binary interactions
- A-A : counting directed interactions
E <- rbind(c(1,2),c(2,2),c(1,1))
typeInter <- c( "inc","diradj", "adj")
v_distrib <- c('ZIgaussian','bernoulli','poisson')
Note that the distributions may be Bernoulli, Poisson, Gaussian or Laplace (with null mean). For the Gaussian distribution, a mean and a variance must be given.
We generate randomly the emission parameters \(\theta\).
list_theta <- list()
list_theta[[1]] <- list()
list_theta[[1]]$mean <- matrix(c(6.1, 8.9, 6.6, 9.8, 2.6, 1.0), 3, 2)
list_theta[[1]]$var <- matrix(c(1.6, 1.6, 1.8, 1.7 ,2.3, 1.5),3, 2)
list_theta[[1]]$p0 <- matrix(c(0.4, 0.1, 0.6, 0.5 , 0.2, 0),3, 2)
list_theta[[2]] <- matrix(c(0.7,1.0, 0.4, 0.6),2, 2)
m3 <- matrix(c(2.5, 2.6 ,2.2 ,2.2, 2.7 ,3.0 ,3.6, 3.5, 3.3),3,3 )
list_theta[[3]] <- (m3 + t(m3))/2# for symetrisation
We are now ready to simulate the data
library(GREMLINS)
dataSim <- rMBM(v_NQ,E , typeInter, v_distrib, list_pi,
list_theta, namesFG = namesFG, seed = 4,keepClassif = TRUE)
list_Net <- dataSim$list_Net
length(list_Net)
#> [1] 3
names(list_Net[[1]])
#> [1] "mat" "typeInter" "rowFG" "colFG"
list_Net[[1]]$typeInter
#> [1] "inc"
list_Net[[1]]$rowFG
#> [1] "A"
list_Net[[1]]$colFG
#> [1] "B"
The model selection and the estimation are performed with the function multipartiteBM.
res_MBMsimu <- multipartiteBM(list_Net,
v_distrib = v_distrib,
namesFG = c('A','B'),
v_Kinit = c(2,2),
nbCores = 2,
initBM = FALSE,
keep = FALSE)
#> [1] "------------Nb of entities in each functional group--------------"
#> A B
#> 60 50
#> [1] "------------Probability distributions on each network--------------"
#> [1] "ZIgaussian" "bernoulli" "poisson"
#> [1] "-------------------------------------------------------------------"
#> [1] " ------ Searching the numbers of blocks starting from [ 2 2 ] blocks"
#> [1] "ICL : -7085.81 . Nb of blocks: [ 2 2 ]"
#> [1] "ICL : -5901.15 . Nb of blocks: [ 3 2 ]"
#> [1] "Best model------ ICL : -5901.15 . Nb of clusters: [ 3 2 ] for [ A , B ] respectively"
We can now get the estimated parameters.
res_MBMsimu$fittedModel[[1]]$paramEstim$list_theta$AB$mean
#> [,1] [,2]
#> [1,] 1.004152 6.572955
#> [2,] 2.582062 8.881842
#> [3,] 9.994673 6.139221
extractClustersMBM produces the clusters in each functional group.
Cl <- extractClustersMBM(res_MBMsimu)
One may also want to estimate the parameters for given numbers of clusters. The function multipartiteBMFixedModel is designed for this task.
res_MBMsimu_fixed <- multipartiteBMFixedModel(list_Net, v_distrib = v_distrib, nbCores = 2,namesFG = namesFG, v_K = c(3,2))
#> [1] "====================== First Forward Step =================="
#> [1] "====================== First Backward Step =================="
#> [1] "====================== Last Forward Step =================="
#> [1] "====================== Last Backward Step =================="
res_MBMsimu_fixed$fittedModel[[1]]$paramEstim$v_K
#> [1] 3 2
extractClustersMBM(res_MBMsimu_fixed)$A
#> [[1]]
#> [1] 1 4 5 10 11 13 15 16 17 23 24 25 27 29 32 33 34 35 39 40 42 48 51 56 57
#> [26] 58 59 60
#>
#> [[2]]
#> [1] 2 6 7 8 12 14 19 22 26 31 36 37 38 41 43 44 46 47 49 50 52
#>
#> [[3]]
#> [1] 3 9 18 20 21 28 30 45 53 54 55
GREMLINS is also able to handle missing data. In the following experiment, we artificially set missing data in the previously simulated matrices.
############# NA data at random in any matrix
epsilon = 10/100
list_Net_NA <- list_Net
for (m in 1:nrow(E)){
U <- sample(c(1,0),v_NQ[E[m,1]]*v_NQ[E[m,2]],replace=TRUE,prob = c(epsilon, 1-epsilon))
matNA <- matrix(U,v_NQ[E[m,1]],v_NQ[E[m,2]])
list_Net_NA[[m]]$mat[matNA== 1] = NA
if (list_Net_NA[[m]]$typeInter == 'adj') {
M <- list_Net_NA[[m]]$mat
diag(M) <- NA
M[lower.tri(M)] = t(M)[lower.tri(M)]
list_Net_NA[[m]]$mat <- M
}
}
res_MBMsimuNA <- multipartiteBM(list_Net_NA,
v_distrib = v_distrib,
namesFG = c('A','B'),
v_Kinit = c(2,2),
nbCores = 2,
keep = FALSE)
#> [1] "------------Nb of entities in each functional group--------------"
#> A B
#> 60 50
#> [1] "------------Probability distributions on each network--------------"
#> [1] "ZIgaussian" "bernoulli" "poisson"
#> [1] "-------------------------------------------------------------------"
#> [1] " ------ Searching the numbers of blocks starting from [ 2 2 ] blocks"
#> [1] "ICL : -6521.28 . Nb of blocks: [ 2 2 ]"
#> [1] "ICL : -5446.97 . Nb of blocks: [ 3 2 ]"
#> [1] " ------ Searching the numbers of blocks starting from [ 3 2 ] blocks"
#> [1] "ICL : -5446.97 . Nb of blocks: [ 3 2 ]"
#> [1] " ------ Searching the numbers of blocks starting from [ 1 2 ] blocks"
#> [1] "ICL : -6977.56 . Nb of blocks: [ 1 2 ]"
#> [1] "ICL : -6521.28 . Nb of blocks: [ 2 2 ]"
#> [1] "Best model------ ICL : -5446.97 . Nb of clusters: [ 3 2 ] for [ A , B ] respectively"
We then have a function to predict the missing edges (probability if binary or intensity if weighted)
pred <- predictMBM(res_MBMsimuNA)