Exploring the latent space of LSMs

• With no node-level covariates, the latent space portion of an LSM is just a low-dimensional representation of the network
  • appropriate as a layout algorithm or for uncovering "clusters" (Krivitsky et al. 2009).

• When node-covariates are of interest, latent space is a "random effect"
  • influence of node covariates on the network are interpreted as an effect on dyadic formation
• The latent space represents both unobserved node-level covariates and unexplained network effects.
• Statistically, LSM are appealing- push out increasingly unexplainable effects to the latent social space
• But what is the qualitative meaning of the latent space in this case?
  • Particularly important as LSMs are sold as a way to visualize data

• how does the latent space inform us of latent dyadic and network-level effects that drove the formation of the network?

• when no effects exist outside observed node-level covariates, what does the latent space look like?

• how do these two things change with different parameterizations of the model?

Overview of approach

• Generate some networks

• Run LSM on them with different parameterizations

• Visualize the latent space, try and grasp qualitative meaning

• Check the latent space for (non) uniformity where we expect it (semi-qualitatively)

\[\begin{align*} \text{logit}\, P[Y_{ij} = 1 ] &= \beta_k^TX_{ijk} - |Z_i-Z_j|, k = 1...\color{red}K \\ \beta_k &\overset{\text{iid}}{\sim} \text{N}(\xi_k,\psi^2_k) \\ Z_i &\overset{\text{iid}}{\sim} \sum_{g=1}^{\color{red}G} \lambda_g\text{MVN}_d(\mu_g,\sigma_g^2I_d) \\ \mu_g &\overset{\text{iid}}{\sim} \text{MVN}_d(0,\omega^2I_d) \\ \sigma_g^2 &\overset{\text{iid}}{\sim} \sigma_0^2\text{Inv}\chi_\alpha^2 \\ (\lambda_1,...,\lambda_g) &\overset{\text{iid}}{\sim} \text{Dirichlet}(\nu_1,...,\nu_g)\end{align*}\]

• Overview

  • Calculate a similarity matrix w/ a baseline tie probability
  • Augment these probabilities with shared group information
  • Draw a random network from the similarity matrix

• Some default parameters:

N_ACTORS <- 8
OUTGROUP_TIE <- 0.01
INGROUP_TIE <- 1
N_COVARIATES <- 1
N_GROUPS <- 2

similarity_matrix <- matrix(0, nrow = N_ACTORS, ncol = N_ACTORS)
similarity_matrix[upper.tri(similarity_matrix)] <- runif(N_ACTORS * (N_ACTORS - 
    1)/2, 0, OUTGROUP_TIE * 2)
pandoc.table(similarity_matrix, style = "rmarkdown", digits = 2)

## Get 'random'' groupings
groupings <- data.frame(id = 1:N_ACTORS, Group1 = c(rep(1, N_ACTORS/2), rep(2, 
    N_ACTORS/2)))
pandoc.table(groupings, style = "rmarkdown", digits = 2)

## Determine matrix of co-memberships in groups, normalized by number of
## groups
percent_shared_memberships <- ifelse(outer(groupings[, 2], groupings[, 2], FUN = "-") == 
    0, 1, 0)
pandoc.table(percent_shared_memberships, style = "rmarkdown", digits = 2)

## Add to si
similarity_matrix <- similarity_matrix + INGROUP_TIE * percent_shared_memberships
diag(similarity_matrix) <- 0
similarity_matrix[lower.tri(similarity_matrix)] <- 0
similarity_matrix <- ifelse(similarity_matrix > 1, 1, similarity_matrix)
pandoc.table(similarity_matrix, style = "rmarkdown", digits = 2)

random_draw <- matrix(0, nrow = N_ACTORS, ncol = N_ACTORS)
random_draw[upper.tri(random_draw)] <- rbinom(rep(1, N_ACTORS * (N_ACTORS - 
    1)/2), 1, as.vector(similarity_matrix[upper.tri(similarity_matrix)]))
lower_indices <- lower.tri(random_draw)
random_draw[lower_indices] <- t(random_draw)[lower_indices]
pandoc.table(random_draw, style = "rmarkdown", digits = 2)

• CIDnetworks vs latentnet packages
• Visual comparison of latent spaces
• Check Ripley's L, a common spatial statistic, for uniformity in the latent space where we (don't) expect it \[\begin{align*} \hat{K}(t) &= \lambda^{-1}\sum_{i \neq j}\frac{I(d_{ij} \, < \, t )}{n} \\ \hat{L}(t) &= (\frac{\hat{K}}{\pi}) \end{align*}\]
  • Do these latter two for different \(NG\), \(K\) and \(G\)

N_ACTORS <- 50
OUTGROUP_TIE <- 0.03
N_COVARIATES <- 1
N_GROUPS <- 2
INGROUP_TIE <- 0.9
N_GAUSSIANS <- 1

N_COVARIATES <- 3

N_GAUSSIANS <- 3

N_COVARIATES <- 2
N_GROUPS <- 4
N_GAUSSIANS <- 1

N_GAUSSIANS <- 2

• Run an LSM on each combination of these parameters
• Get some kind of point estimate of \(\hat{L}(t)\)

• "Findings"
  • The latent space does give some inclination as to dyadic covariates missing from the model
  • When the latent space is representing only noise, it appears close to a random process along Ripley's L
  • Neither of these things appear to change much with a "mis-parameterized" LSM
• None of this is particularly surprising. But...
  • In playing around, I noticed that as OUTGROUP_TIE decreased, these results deteriorate
  • Thus, it would be interesting to work with more realistic networks and see if these results hold

• Pavel Krivitsky, Mark Handcock, Adrian Raftery, Peter Hoff, (2009) Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models. Social Networks 31 (3) 204-213 10.1016/j.socnet.2009.04.001