Simulate networks from a fitted AME model

Generates multiple network realizations from the posterior distribution of a fitted AME model. This function performs posterior predictive simulation by drawing from the full joint posterior distribution of model parameters, thereby propagating parameter uncertainty into the simulated networks.

Usage

# S3 method for class 'ame'
simulate(
  object,
  nsim = 100,
  seed = NULL,
  newdata = NULL,
  burn_in = 0,
  thin = 1,
  return_latent = FALSE,
  ...
)

Arguments

object

fitted model object of class "ame"

nsim

number of networks to simulate (default: 100)

seed

random seed for reproducibility

newdata

optional list containing new covariate data:

Xdyad: dyadic covariates (n x n x p array or nA x nB x p for bipartite)
Xrow: row/sender covariates (n x p matrix or nA x p for bipartite)
Xcol: column/receiver covariates (n x p matrix or nB x p for bipartite)

If NULL, uses covariates from original model fit

burn_in

number of initial MCMC samples to discard (default: 0, assumes burn-in already removed)

thin

thinning interval for MCMC samples (default: 1, use every sample)

return_latent

logical: return latent Z matrices in addition to Y? (default: FALSE)

...

additional arguments (not currently used)

Value

A list with components:

Y: list of nsim simulated networks in the same format as the original data
Z: if return_latent=TRUE, list of nsim latent Z matrices
family: the family of the model (binary, normal, etc.)
mode: network mode (unipartite or bipartite)

Details

Mathematical Framework:

The AME model represents networks through a latent variable framework: $$Y_{ij} \sim F(Z_{ij})$$ where F is the observation model (e.g., probit for binary) and Z is the latent network: $$Z_{ij} = \beta^T x_{ij} + a_i + b_j + u_i^T v_j + \epsilon_{ij}$$

Components:

$\beta$: regression coefficients for dyadic/nodal covariates
$a_i, b_j$: additive sender and receiver random effects
$u_i, v_j$: multiplicative latent factors (dimension R)
$\epsilon_{ij}$: dyadic random effects with correlation $\rho$

Uncertainty Quantification Process:

For each simulated network k = 1, ..., nsim:

Parameter Sampling: Draw parameter set $\theta^{(k)}$ from MCMC chains:
- Sample iteration s uniformly from stored MCMC samples
- Extract $\beta^{(s)}$, variance components $(v_a^{(s)}, v_b^{(s)}, v_e^{(s)}, \rho^{(s)})$
Random Effects Generation: Sample new random effects from posterior distributions:
- $a_i^{(k)} \sim N(0, v_a^{(s)})$ for i = 1, ..., n (row effects)
- $b_j^{(k)} \sim N(0, v_b^{(s)})$ for j = 1, ..., m (column effects)
- Note: We sample fresh from the posterior variance rather than using point estimates to properly propagate uncertainty
Latent Network Construction: Build expected latent positions: $$E[Z_{ij}^{(k)}] = \beta^{(s)T} x_{ij} + a_i^{(k)} + b_j^{(k)} + \hat{u}_i^T \hat{v}_j$$ where $\hat{u}_i, \hat{v}_j$ are posterior mean latent factors
Dyadic Correlation: Add correlated noise structure: $$Z_{ij}^{(k)} = E[Z_{ij}^{(k)}] + \epsilon_{ij}^{(k)}$$ where $\epsilon$ has covariance structure: $$Cov(\epsilon_{ij}, \epsilon_{ji}) = \rho^{(s)} v_e^{(s)}$$ $$Var(\epsilon_{ij}) = v_e^{(s)}$$
Observation Model: Generate observed network based on family:
- Binary: $Y_{ij}^{(k)} = I(Z_{ij}^{(k)} > 0)$
- Normal: $Y_{ij}^{(k)} = Z_{ij}^{(k)}$
- Poisson: $Y_{ij}^{(k)} \sim Poisson(\exp(Z_{ij}^{(k)}))$
- Other families use appropriate link functions

Sources of Uncertainty:

The simulation captures three types of uncertainty:

Parameter uncertainty: Different MCMC samples yield different $\beta, v_a, v_b, v_e, \rho$
Random effect uncertainty: Fresh draws from $N(0, v_a), N(0, v_b)$ for each simulation
Dyadic uncertainty: Correlated random noise $\epsilon_{ij}$

This approach provides proper posterior predictive distributions that account for all sources of uncertainty in the model. The variation across simulated networks reflects our posterior uncertainty about the data generating process.

Limitations:

Currently, multiplicative effects (U, V) use posterior means rather than sampling from their full posterior. For complete uncertainty quantification, one would need to store and sample from the full MCMC chains of these latent factors, which would require substantial additional memory.

Symmetric Networks:

For symmetric networks, the model enforces $a_i = b_i$ and $u_i = v_i$, and the latent matrix Z is symmetrized before generating observations.

Author

Shahryar Minhas

Examples

if (FALSE) { # \dontrun{
# Fit a model
data(YX_bin)
fit <- ame(YX_bin$Y, YX_bin$X, burn=100, nscan=500, family="binary")

# Simulate 50 networks from posterior
sims <- simulate(fit, nsim=50)

# With new covariates
new_X <- array(rnorm(dim(YX_bin$X)[1] * dim(YX_bin$X)[2] * dim(YX_bin$X)[3]),
               dim=dim(YX_bin$X))
sims_new <- simulate(fit, nsim=50, newdata=list(Xdyad=new_X))
} # }