This is an R package implementation of a Stochastic Block Model for Multilevel Network (MLVSBM) as described in Chabert-Liddell, Barbillon, Donnet and Lazega (2021) https://doi.org/10.1016/j.csda.2021.107179.
You can install the latest released version of MLVSBM from cran
install.packages("MLVSBM")or the latest dev version from github with:
devtools::install_github("Chabert-Liddell/MLVSBM")The package deals with multilevel network defined as the junction of two interaction networks (adjacency matrices) linked by an affiliation relationship (affiliation matrix).
First, we’re going to simulate a multilevel network with 100 individuals and 3 clusters of individuals for the lower level and 50 organizations and 3 clusters for the upper level. The inter-organizational level will have an assortative structure and will be undirected, the inter-individual’s one a core-periphery structure and will be directed. Affiliation matrix will be generated by a power law and the dependence between the structure of the two levels will be strong.
set.seed(123)
my_mlvsbm <- MLVSBM::mlvsbm_simulate_network(
n = list(I = 60, O = 40), # Number of nodes for the lower level and the upper level
Q = list(I = 3, O = 3), # Number of blocks for the lower level and the upper level
pi = c(.5, .3, .2), # Block proportion for the upper level, must sum to one
gamma = matrix(c(.8, .1, .1, # Block proportion for the lower level,
.1, .8, .1,
.1, .1, .8), # each column must sum to one
nrow = 3, ncol = 3, byrow = TRUE),
alpha = list(I = matrix(c(.1, .1, .3,
.1, .2, .5,
.1, .5, .5),
nrow = 3, ncol = 3, byrow = TRUE), # Connection matrix
O = matrix(c(.5, .1, .1,
.1, .5, .1,
.1, .1, .5),
nrow = 3, ncol = 3, byrow = TRUE)),# between blocks
directed = list(I = TRUE, O = FALSE), # Are the upper and lower level directed or not ?
affiliation = "preferential", # How the affiliation matrix is generated
no_empty_org = FALSE) # May the affiliation matrix have column suming to 0The network is stocked in an R6 object of type
MLVSBM.
Now, we are going to create a multilevel network object from 2 existing adjacency matrix and an affiliation matrix :
lower_level <- my_mlvsbm$adjacency_matrix$I # matrix of size nI x nI
upper_level <- my_mlvsbm$adjacency_matrix$O # matrix of size nO x nO
affiliation <- my_mlvsbm$affiliation_matrix # matrix of size nI x nO
my_mlvsbm2 <- MLVSBM::mlvsbm_create_network(X = list(I = lower_level, O = upper_level),
A = affiliation)We can now infer the parameters, blocks and edge probabilities of our
network by using the mlvlsbm_estimate_network() function on
an MLVSBM object. It will return the best model for this
network as another R6 object of type FitMLVSBM.
fit <- MLVSBM:::mlvsbm_estimate_network(my_mlvsbm)
#>
#> [1] "Infering lower level :"
#> [1] "# blocks: 3, ICL = -1753.45290053552 !"
#>
#> [1] "Infering upper level :"
#> [1] "# blocks: 3, ICL = -429.333753035444 !"
#> [1] "======= # Individual clusters : 3 , # Organisation clusters 3, ICL : -2161.44453717678========"
#> [1] "======= # Individual blocks : 3 , # Organizational blocks 3, ICL : -2161.44453717678========"
#> [1] "ICL for independent levels : -2182.78665357096"
#> [1] "ICL for interdependent levels : -2161.44453717678"
#> [1] "=====Interdependence is detected between the two levels!====="Some generic functions allow an easy access to the most useful parameters:
print(fit)
#> Multilevel Stochastic Block Model -- bernoulli variant
#> =====================================================================
#> Dimension = ( 60 40 ) - ( 3 3 ) blocks.
#> =====================================================================
#> * Useful fields
#> $independent, $distribution, $nb_nodes, $nb_clusters, $Z
#> $membership, $parameters, $ICL, $vbound, $X_hatWe can also plot a synthetic view of the network, with the fitted model:
plot(fit)
#> Joining, by = c("name", "group")
#> Joining, by = "name"
As well as obtain the model parameters:
coef(fit)
#> $alpha
#> $alpha$I
#> [,1] [,2] [,3]
#> [1,] 0.23072817 0.5382933 0.09444543
#> [2,] 0.49659089 0.4545821 0.08339372
#> [3,] 0.09614086 0.2840693 0.13421325
#>
#> $alpha$O
#> [,1] [,2] [,3]
#> [1,] 0.66670837 0.09845986 0.0557177
#> [2,] 0.09845986 0.48056388 0.1137974
#> [3,] 0.05571770 0.11379738 0.5299402
#>
#>
#> $pi
#> $pi$O
#> [1] 0.1499768 0.5496535 0.3003697
#>
#>
#> $gamma
#> [,1] [,2] [,3]
#> [1,] 0.09064248 0.19243672 8.692712e-01
#> [2,] 0.72734754 0.03843131 1.307266e-01
#> [3,] 0.18200998 0.76913196 2.105228e-06And the prediction of the block clustering as well as the probability of a link between two individuals or two organizations:
pred <- predict(fit)
pred$nodes
#> $I
#> I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17 I18 I19 I20
#> 1 3 3 1 3 2 1 2 1 3 1 3 3 1 3 3 1 1 3 1
#> I21 I22 I23 I24 I25 I26 I27 I28 I29 I30 I31 I32 I33 I34 I35 I36 I37 I38 I39 I40
#> 1 1 3 1 3 3 1 1 1 1 3 1 2 2 1 3 2 3 3 3
#> I41 I42 I43 I44 I45 I46 I47 I48 I49 I50 I51 I52 I53 I54 I55 I56 I57 I58 I59 I60
#> 2 1 1 1 2 2 1 3 3 1 2 1 3 1 2 2 2 1 3 3
#>
#> $O
#> O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12 O13 O14 O15 O16 O17 O18 O19 O20
#> 3 2 2 2 1 2 1 1 3 2 3 3 3 2 2 2 2 3 2 2
#> O21 O22 O23 O24 O25 O26 O27 O28 O29 O30 O31 O32 O33 O34 O35 O36 O37 O38 O39 O40
#> 2 3 2 3 2 2 1 1 1 2 2 3 2 3 2 2 3 2 2 3
pred$dyads$I[1:5, 1:5]
#> I1 I2 I3 I4 I5
#> I1 0.00000000 0.09444600 0.09444600 0.23073017 0.09444714
#> I2 0.09614226 0.00000000 0.13421327 0.09614210 0.13421295
#> I3 0.09614226 0.13421327 0.00000000 0.09614210 0.13421295
#> I4 0.23073020 0.09444601 0.09444601 0.00000000 0.09444714
#> I5 0.09614339 0.13421294 0.13421294 0.09614322 0.00000000Output of the algorithm are stocked in the MLVSBM and
FitMLVSBM objects. The MLVSBM object stocks
information of the observed or simulated network and a list of all the
fitted SBM and MLVSBM models.
my_mlvsbm$ICL # A data frame of the inferred models
#> index Q_I Q_O ICL
#> 1 1 3 3 -2161.445
my_fit <- my_mlvsbm$fittedmodels[[which.max(my_mlvsbm$ICL$ICL)]] # The fitted model with index the highest ICL
my_mlvsbm$ICL_sbm # The ICL of the SBM
#> $lower
#> [1] -Inf -Inf -1753.453 -Inf -Inf -Inf -Inf
#> [8] -Inf -Inf -Inf
#>
#> $upper
#> [1] -Inf -Inf -429.3338 -Inf -Inf -Inf -Inf
#> [8] -Inf -Inf -Inf
my_sbm_lower <- my_mlvsbm$fittedmodels_sbm$lower[[3]] # A fitted SBM for the lower level with 3 blocks
my_sbm_upper <- my_mlvsbm$fittedmodels_sbm$upper[[2]] # A fitted SBM for the upper level with 2 blocksYou can also get the parameters and the clustering of the fitted
model from the FitMLVSBM object as follows:
fit$parameters # The connectivity and membership parameters of the model
#> $alpha
#> $alpha$I
#> [,1] [,2] [,3]
#> [1,] 0.23072817 0.5382933 0.09444543
#> [2,] 0.49659089 0.4545821 0.08339372
#> [3,] 0.09614086 0.2840693 0.13421325
#>
#> $alpha$O
#> [,1] [,2] [,3]
#> [1,] 0.66670837 0.09845986 0.0557177
#> [2,] 0.09845986 0.48056388 0.1137974
#> [3,] 0.05571770 0.11379738 0.5299402
#>
#>
#> $pi
#> $pi$O
#> [1] 0.1499768 0.5496535 0.3003697
#>
#>
#> $gamma
#> [,1] [,2] [,3]
#> [1,] 0.09064248 0.19243672 8.692712e-01
#> [2,] 0.72734754 0.03843131 1.307266e-01
#> [3,] 0.18200998 0.76913196 2.105228e-06
fit$Z # The membership of each nodes
#> $I
#> I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17 I18 I19 I20
#> 1 3 3 1 3 2 1 2 1 3 1 3 3 1 3 3 1 1 3 1
#> I21 I22 I23 I24 I25 I26 I27 I28 I29 I30 I31 I32 I33 I34 I35 I36 I37 I38 I39 I40
#> 1 1 3 1 3 3 1 1 1 1 3 1 2 2 1 3 2 3 3 3
#> I41 I42 I43 I44 I45 I46 I47 I48 I49 I50 I51 I52 I53 I54 I55 I56 I57 I58 I59 I60
#> 2 1 1 1 2 2 1 3 3 1 2 1 3 1 2 2 2 1 3 3
#>
#> $O
#> O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12 O13 O14 O15 O16 O17 O18 O19 O20
#> 3 2 2 2 1 2 1 1 3 2 3 3 3 2 2 2 2 3 2 2
#> O21 O22 O23 O24 O25 O26 O27 O28 O29 O30 O31 O32 O33 O34 O35 O36 O37 O38 O39 O40
#> 2 3 2 3 2 2 1 1 1 2 2 3 2 3 2 2 3 2 2 3
fit$vbound # A vector of the varational bound of the VEM algorithm
#> [1] -2088.490 -2088.447 -2088.447 -2088.447
tau <- fit$membership # The variational parameters of the model
pred <- fit$X_hat # The links predictions for each level