Generates multiple longitudinal network realizations from the posterior distribution of a fitted LAME (Longitudinal AME) model. This function performs posterior predictive simulation for dynamic networks, propagating both cross-sectional and temporal uncertainty through the simulated trajectories.
Usage
# S3 method for class 'lame'
simulate(
object,
nsim = 100,
seed = NULL,
newdata = NULL,
n_time = NULL,
burn_in = 0,
thin = 1,
return_latent = FALSE,
start_from = "posterior",
...
)Arguments
- object
fitted model object of class "lame"
- nsim
number of network trajectories to simulate (default: 100)
- seed
random seed for reproducibility
- newdata
optional list containing new covariate data:
- Xdyad
list of T dyadic covariate arrays (n x n x p or nA x nB x p)
- Xrow
list of T row/sender covariate matrices (n x p or nA x p)
- Xcol
list of T column/receiver covariate matrices (n x p or nB x p)
If NULL, uses covariates from original model fit
- n_time
number of time periods to simulate. If NULL, uses same as original data
- burn_in
number of initial MCMC samples to discard (default: 0)
- thin
thinning interval for MCMC samples (default: 1, use every sample)
- return_latent
logical: return latent Z matrices in addition to Y? (default: FALSE)
- start_from
character: how to initialize the simulation
- "posterior"
start from posterior mean (default)
- "random"
random initialization
- "data"
use first time point from original data
- ...
additional arguments (not currently used)
Value
A list with components:
- Y
list of nsim simulated longitudinal network trajectories, each element is a list of T networks
- Z
if return_latent=TRUE, list of nsim latent Z trajectories
- family
the family of the model (binary, normal, etc.)
- mode
network mode (unipartite or bipartite)
- n_time
number of time periods
Details
Mathematical Framework for Longitudinal Networks:
The LAME model extends AME to multiple time periods T with potential temporal dependencies. For each time t = 1, ..., T: $$Y_{ij,t} \sim F(Z_{ij,t})$$ where the latent network evolves as: $$Z_{ij,t} = \beta^T x_{ij,t} + a_{i,t} + b_{j,t} + u_{i,t}^T v_{j,t} + \epsilon_{ij,t}$$
Temporal Dynamics:
LAME can incorporate three types of temporal dependencies:
Static Effects: Parameters constant over time
\(a_{i,t} = a_i\), \(b_{j,t} = b_j\) for all t
\(u_{i,t} = u_i\), \(v_{j,t} = v_j\) for all t
Dynamic Additive Effects: AR(1) process for random effects $$a_{i,t} = \rho_{ab} a_{i,t-1} + \eta_{i,t}, \quad \eta_{i,t} \sim N(0, \sigma_a^2(1-\rho_{ab}^2))$$ $$b_{j,t} = \rho_{ab} b_{j,t-1} + \xi_{j,t}, \quad \xi_{j,t} \sim N(0, \sigma_b^2(1-\rho_{ab}^2))$$ where \(\rho_{ab}\) is the temporal correlation parameter
Dynamic Multiplicative Effects: AR(1) for latent factors $$u_{i,t} = \rho_{uv} u_{i,t-1} + \omega_{i,t}$$ $$v_{j,t} = \rho_{uv} v_{j,t-1} + \psi_{j,t}$$
Uncertainty Quantification Process for Trajectories:
For each simulated trajectory k = 1, ..., nsim:
Step 1: Parameter Sampling
Draw MCMC iteration s uniformly from stored posterior samples
Extract static parameters: \(\beta^{(s)}\), variance components
Extract temporal parameters if applicable: \(\rho_{ab}^{(s)}\), \(\rho_{uv}^{(s)}\)
Step 2: Initialize at t = 1
Depending on start_from parameter:
"posterior": Use posterior means as starting values
"random": Draw from stationary distribution
For additive effects: \(a_{i,1}^{(k)} \sim N(0, \sigma_a^2)\)
For multiplicative effects: Initialize from prior
Step 3: Evolve Through Time
For each t = 2, ..., T:
a) Update Dynamic Effects (if applicable): $$a_{i,t}^{(k)} = \rho_{ab}^{(s)} a_{i,t-1}^{(k)} + \eta_{i,t}^{(k)}$$ where \(\eta_{i,t}^{(k)} \sim N(0, \sigma_a^2(1-[\rho_{ab}^{(s)}]^2))\)
The innovation variance \(\sigma_a^2(1-\rho_{ab}^2)\) ensures stationarity
b) Construct Latent Network: $$E[Z_{ij,t}^{(k)}] = \beta^{(s)T} x_{ij,t} + a_{i,t}^{(k)} + b_{j,t}^{(k)} + u_{i,t}^T v_{j,t}$$
c) Add Dyadic Noise: $$Z_{ij,t}^{(k)} = E[Z_{ij,t}^{(k)}] + \epsilon_{ij,t}^{(k)}$$ with correlation structure preserved from AME model
d) Generate Observations: Apply appropriate link function based on family
Sources of Uncertainty in Longitudinal Context:
Cross-sectional uncertainty (as in AME):
Parameter uncertainty from MCMC
Random effect variability
Dyadic noise
Temporal uncertainty:
Uncertainty in temporal correlation parameters \(\rho_{ab}, \rho_{uv}\)
Innovation noise in AR(1) processes
Propagation of uncertainty through time (compounds over periods)
Initial condition uncertainty:
Different starting values lead to different trajectories
Captured through start_from options
Interpretation of Multiple Trajectories:
Each simulated trajectory represents one possible evolution of the network consistent with the posterior distribution. Variation across trajectories captures:
Model parameter uncertainty
Stochastic variation in temporal evolution
Accumulated uncertainty over time periods
The ensemble of trajectories provides prediction intervals that widen over time, reflecting increasing uncertainty in longer-term forecasts.
Special Considerations:
Temporal Correlation: Higher \(\rho\) values create smoother trajectories with more persistence
Stationarity: The AR(1) innovation variance is scaled to maintain stationary marginal distributions
Missing Time Points: If simulating beyond observed data (n_time > T_observed), covariates are recycled or set to zero with appropriate warnings
Limitations:
As with simulate.ame, multiplicative effects currently use posterior means. Full uncertainty would require storing complete MCMC chains for \(u_{i,t}, v_{j,t}\) at all time points, which is memory-intensive for large networks and long time series.
Examples
if (FALSE) { # \dontrun{
# Fit a longitudinal model
data(YX_bin_list)
fit <- lame(YX_bin_list$Y, YX_bin_list$X, burn=100, nscan=500, family="binary")
# Simulate 50 network trajectories from posterior
sims <- simulate(fit, nsim=50)
# Simulate longer time series
sims_long <- simulate(fit, nsim=25, n_time=20)
# With new covariates
new_X <- lapply(YX_bin_list$X, function(x) {
array(rnorm(prod(dim(x))), dim=dim(x))
})
sims_new <- simulate(fit, nsim=50, newdata=list(Xdyad=new_X))
} # }