# MLTA - Code for Gollini, I. (in press) - A mixture model approach for clustering bipartite networks

#### 12/06/2019

This code accompany the paper Gollini, I. (in press) “A mixture model approach for clustering bipartite networks”, Challenges in Social Network Research Volume in the Lecture Notes in Social Networks (LNSN - Series of Springer). Preprint: arXiv:1905.02659.

set.seed(123)

Load the noordin dataset which is available in the manet package.

data(noordin, package = "manet")

Load the lvm4net package and set the incidence matrix in the correct format, and give names to the sender nodes.

library(lvm4net)
X <- as.matrix(noordin)
namesX <- paste('V', seq(1, nrow(X)))

Heatmap of the data:

heatmap(
X,
Rowv = NA,
Colv = NA,
col = grey(c(0.95, 0.0)),
scale = "none",
margins = c(3, 3),
xlab = "Event",
ylab = "Terrorist"
)

We want to fit the model on a range of groups G, from 2 to 4 and the latent continuous latent variable takes value D from 0 to 3.

G <- 2:4 # is the number of groups
D <- 0:3 # is the dimension of the latent variable

Fit the mlta

mod.mlta <- mlta(X, G = G, D = D, wfix = FALSE) # It takes ~ 2 minutes with 3 starts
mod.mlta.wfix <- mlta(X, G = G, D = 1:3, wfix = TRUE) # It takes ~ 2 minutes with 3 starts
mod.mlta$BIC$Table of BIC Results
##              G=2      G=3      G=4
## dim y=0 2087.159 2187.163 2305.239
## dim y=1 2165.918 2412.358 2732.959
## dim y=2 2402.889 2885.561 3412.363
## dim y=3 2792.768 3409.382 4154.685
mod.mlta.wfix$BIC$Table of BIC Results
##              G=2      G=3      G=4
## dim y=1 2027.425 2126.788 2300.028
## dim y=2 2137.959 2254.079 2398.944
## dim y=3 2317.978 2446.674 2569.771

According to the BIC the best model selected is has two groups (G = 2) and a one dimensional continuous latent variable (D = 1) and common slope parameters across groups (wfix = TRUE).

res <- mod.mlta.wfix[[1]]
plot(c(res$w), xlab = "Event", ylab = "w", pch = 19) abline(h = 0) par(mfrow = c(1, 2)) plot(c(res$b[1,]), xlab = "Event", ylab = "b", pch = 19, main = "Group 1")
abline(h = 0)

plot(c(res$b[2,]), xlab = "Event", ylab = "b", pch = 19, main = "Group 2") abline(h = 0) Plot the probability of each sender node to belong to group 1. plot(res$z[,1], pch = 19,
xlab = "Sender node",
ylab = "Probability to belong to group 1")
abline(h = 0.5, col = "red")

Find and plot the probability that the median sender node in group g has a link with receiver node r.

pig0 <- 1 / ( 1 + exp(-res\$b))

matplot(t(pig0), type = "l",
ylim = c(0, 1), ylab = expression(paste(pi[rg](0))),
xlab = "Receiver node (r)", xaxt = "n",
main = "Probability that the median sender node in group g\n has a link with receiver node r")
axis(1, at = 1:ncol(X))
legend("topright", paste("Group", 1:2, sep = " "), col = 1:2, lty = 1:2)

We can calculate the log-lift for the best model selected.

loglift <- log(lift(res, pdGH = 21))
heatmap(
loglift[,,1],
Rowv = NA,
Colv = NA,
col = colorspace::diverge_hsv(20),
breaks = seq(-10, 10, by = 1),
revC = TRUE,
scale = "none",
xlab = "Event",
ylab = "Event",
main = "Log-Lift for Group 1"
)

heatmap(
loglift[,,2],
Rowv = NA,
Colv = NA,
col = colorspace::diverge_hsv(20),
breaks = seq(-10, 10, by = 1),
revC = TRUE,
scale = "none",
xlab = "Event",
ylab = "Event",
main = "Log-Lift for Group 2"
)

### Extra code:

When D = 0 the mlta reduces to the latent class analysis that can be fitted by using lca

mod.lca <- lca(X, G = 2:4)

When there are no groups (G = 1) and D > 0 the mlta reduces to the latent trait analysis that can be fitted by using lta

mod.lta <- lta(X, D = 1:3)