An MCMC routine providing a fit to an additive and multiplicative effects
(AME) regression model to longitudinal (time-series) relational data of
various types. Supports both unipartite (square) and bipartite (rectangular)
network structures. For cross-sectional (single time point) networks,
use the ame function.
Usage
lame(Y,Xdyad=NULL, Xrow=NULL, Xcol=NULL, rvar = !(family=="rrl"),
cvar = TRUE, dcor = !symmetric, nvar=TRUE, R = 0, R_row = NULL, R_col = NULL,
mode = c("unipartite", "bipartite"),
dynamic_uv = FALSE, dynamic_ab = FALSE, dynamic_G = FALSE, family ="normal",
intercept=!is.element(family,c("rrl","ordinal")),
symmetric=FALSE,
odmax=NULL, prior=list(), g=NA,
seed = 6886, nscan = 10000, burn = 500, odens = 25, plot=FALSE, print = FALSE, gof=TRUE,
start_vals=NULL, periodic_save=FALSE, out_file=NULL, save_interval=0.25, model.name=NULL)Arguments
- Y
a T length list of n x n relational matrices, where T corresponds to the number of replicates (over time, for example). See family below for various data types.
- Xdyad
a T length list of n x n x pd arrays of covariates
- Xrow
a T length list of n x pr matrices of nodal row covariates
- Xcol
a T length list of n x pc matrices of nodal column covariates
- rvar
logical: fit row random effects (asymmetric case)?
- cvar
logical: fit column random effects (asymmetric case)?
- dcor
logical: fit a dyadic correlation (asymmetric case)?
- nvar
logical: fit nodal random effects (symmetric case)?
- R
integer: dimension of the multiplicative effects (can be zero)
- R_row
integer: for bipartite networks, dimension of row node multiplicative effects (defaults to R)
- R_col
integer: for bipartite networks, dimension of column node multiplicative effects (defaults to R)
- mode
character: either "unipartite" (default) for square networks or "bipartite" for rectangular networks
- dynamic_uv
logical: fit dynamic multiplicative effects (latent factors) that evolve over time using AR(1) processes. When TRUE, the latent positions U and V become time-varying, following \(U_{i,t} = \rho_{uv} U_{i,t-1} + \epsilon_{i,t}\), where epsilon follows N(0, sigma_uv^2). This allows actors' positions in latent social space to drift smoothly over time, capturing evolving network structure and community dynamics. Inspired by dynamic latent space models (Sewell and Chen 2015, "Latent Space Models for Dynamic Networks", JASA; Durante and Dunson 2014, "Nonparametric Bayes Dynamic Modeling of Relational Data", Biometrika). The implementation uses efficient blocked Gibbs sampling with C++ acceleration for scalability. Default FALSE.
- dynamic_ab
logical: fit dynamic additive effects (sender/receiver effects) that evolve over time using AR(1) processes. When TRUE, the row effects (a) and column effects (b) become time-varying, following \(a_{i,t} = \rho_{ab} a_{i,t-1} + \epsilon_{i,t}\). This captures temporal heterogeneity in actors' baseline propensities to send and receive ties, allowing for smooth changes in activity levels and popularity over time. For example, an actor's tendency to form outgoing ties might gradually increase or decrease across observation periods. The AR(1) specification ensures temporal smoothness while allowing for actor-specific evolution patterns. Implementation uses conjugate updates where possible and C++ for computational efficiency. Default FALSE.
- dynamic_G
logical: for bipartite networks, fit dynamic interaction matrix G that evolves over time. When TRUE, the rectangular interaction matrix G becomes time-varying, allowing the mapping between row and column latent spaces to change over time. Default FALSE.
- family
character: one of "normal","tobit","binary","ordinal","cbin","frn","rrl","poisson" - see the details below
- intercept
logical: fit model with an intercept?
- symmetric
logical: Is the sociomatrix symmetric by design?
- odmax
a scalar integer or vector of length n giving the maximum number of nominations that each node may make - used for "frn" and "cbin" families
- prior
a list containing hyperparameters for the prior distributions. Available options and their defaults:
- Sab0
Prior covariance matrix for additive effects (default: diag(2)). A 2x2 matrix where Sab0\[1,1\] is the prior variance for row effects, Sab0\[2,2\] is the prior variance for column effects, and off-diagonals control correlation between row and column effects.
- eta0
Prior degrees of freedom for covariance of multiplicative effects (default: 4 + 3 \* n/100, where n is the number of actors). Higher values impose stronger shrinkage on the latent factors.
- etaab
Prior degrees of freedom for covariance of additive effects (default: 4 + 3 \* n/100). Controls shrinkage of row/column random effects.
- rho_uv_mean
For dynamic_uv=TRUE: Prior mean for UV AR(1) parameter (default: 0.9). Values close to 1 indicate high temporal persistence.
- rho_uv_sd
For dynamic_uv=TRUE: Prior SD for UV AR(1) parameter (default: 0.1). Controls uncertainty about temporal dependence.
- sigma_uv_shape
For dynamic_uv=TRUE: Shape parameter for inverse-gamma prior on UV innovation variance (default: 2).
- sigma_uv_scale
For dynamic_uv=TRUE: Scale parameter for inverse-gamma prior on UV innovation variance (default: 1).
- rho_ab_mean
For dynamic_ab=TRUE: Prior mean for additive effects AR(1) parameter (default: 0.8). Controls temporal smoothness of sender/receiver effects.
- rho_ab_sd
For dynamic_ab=TRUE: Prior SD for additive effects AR(1) parameter (default: 0.15).
- sigma_ab_shape
For dynamic_ab=TRUE: Shape parameter for inverse-gamma prior on additive effects innovation variance (default: 2).
- sigma_ab_scale
For dynamic_ab=TRUE: Scale parameter for inverse-gamma prior on additive effects innovation variance (default: 1).
Common usage: prior = list(Sab0 = diag(c(1, 1)), eta0 = 10) for stronger shrinkage, or prior = list(rho_uv_mean = 0.95) for higher temporal persistence.
- g
optional scalar or vector for g-prior on regression coefficients. Default is p^2 where p is the number of regression parameters. The g-prior controls the variance of regression coefficients: larger values allow for larger coefficient values. Can be a vector of length p for parameter-specific control.
- seed
random seed
- nscan
number of iterations of the Markov chain (beyond burn-in)
- burn
burn in for the Markov chain
- odens
output density for the Markov chain
- plot
logical: plot results while running?
logical: print results while running?
- gof
logical: calculate goodness of fit statistics?
- start_vals
List from previous model run containing parameter starting values for new MCMC
- periodic_save
logical: indicating whether to periodically save MCMC results
- out_file
character vector indicating name and path in which file should be stored if periodic_save is selected. For example, on an Apple OS out_file="~/Desktop/ameFit.rda".
- save_interval
quantile interval indicating when to save during post burn-in phase.
- model.name
optional string for model selection output
Value
- BETA
posterior samples of regression coefficients
- VC
posterior samples of the variance parameters
- APM
posterior mean of additive row effects a
- BPM
posterior mean of additive column effects b
- U
posterior mean of multiplicative row effects u. For dynamic_uv=TRUE, this is a 3D array (n x R x T)
- V
posterior mean of multiplicative column effects v (asymmetric case). For dynamic_uv=TRUE, this is a 3D array (n x R x T)
- UVPM
posterior mean of UV
- ULUPM
posterior mean of ULU (symmetric case)
- L
posterior mean of L (symmetric case)
- EZ
estimate of expectation of Z matrix
- YPM
posterior mean of Y (for imputing missing values)
- GOF
observed (first row) and posterior predictive (remaining rows) values of four goodness-of-fit statistics
- start_vals
Final parameter values from MCMC, can be used as the input for a future model run.
- model.name
Name of the model (if provided)
Details
This command provides posterior inference for parameters in AME models of longitudinal relational data, assuming one of eight possible data types/models. The model supports both unipartite networks (square adjacency matrices) and bipartite networks (rectangular adjacency matrices with distinct row and column node sets) across multiple time points.
Dynamic Effects Implementation:
The dynamic_uv and dynamic_ab parameters enable time-varying latent representations through autoregressive processes. These extensions are particularly useful for understanding how network structure evolves over time.
Dynamic Multiplicative Effects (dynamic_uv=TRUE): The latent factors U and V evolve according to AR(1) processes: $$U_{i,k,t} = \rho_{uv} U_{i,k,t-1} + \epsilon_{i,k,t}$$ where \(\epsilon_{i,k,t} \sim N(0, \sigma_{uv}^2)\), i indexes actors, k indexes latent dimensions, and t indexes time. The parameter \(\rho_{uv}\) controls temporal persistence (values near 1 indicate slow evolution). This captures time-varying homophily, latent community structure, and transitivity dynamics.
Key references:
Sewell & Chen (2015): Introduced dynamic latent space models with actor-specific evolution rates
Durante & Dunson (2014): Nonparametric Bayesian approach allowing flexible evolution of network structure
Hoff (2011): Hierarchical multilinear models providing theoretical foundation for temporal dependencies
Dynamic Additive Effects (dynamic_ab=TRUE): The sender (a) and receiver (b) effects evolve as: $$a_{i,t} = \rho_{ab} a_{i,t-1} + \epsilon_{i,t}$$ $$b_{i,t} = \rho_{ab} b_{i,t-1} + \eta_{i,t}$$ where \(\epsilon_{i,t}, \eta_{i,t} \sim N(0, \sigma_{ab}^2)\). This models time-varying individual activity levels (outdegree) and popularity (indegree).
Applications include:
Tracking changes in node centrality over time
Identifying emerging influential actors
Detecting declining activity patterns
Modeling life-cycle effects in social networks
Prior Specification for Dynamic Parameters:
\(\rho_{uv}, \rho_{ab} \sim TruncNormal(mean, sd, 0, 1)\): Ensures stationarity of AR(1) process
\(\sigma_{uv}^2, \sigma_{ab}^2 \sim InverseGamma(shape, scale)\): Controls innovation variance
Default priors (\(\rho_{uv}\) mean=0.9, \(\rho_{ab}\) mean=0.8) favor smooth evolution
Adjust rho_*_mean closer to 1 for slower evolution, closer to 0 for more rapid changes
Computational Considerations:
Dynamic effects increase computation by ~30-50\
Memory usage scales as O(nRT) for dynamic_uv, O(n*T) for dynamic_ab
C++ implementation provides ~70\
Convergence diagnostics: Monitor rho and sigma parameters carefully
Effective sample sizes typically lower due to temporal correlation
Recommend burn >= 1000 and nscan >= 20000 for dynamic models
Model Selection Guidelines: Use both dynamic_uv and dynamic_ab when:
Networks show clear temporal trends in density or clustering
Individual node behavior changes systematically over time
Community structure evolves (merging, splitting, drift)
Use only dynamic_uv when:
Latent structure/communities change but individual effects are stable
Focus is on evolving homophily or clustering patterns
Network shows structural reconfiguration over time
Use only dynamic_ab when:
Individual heterogeneity varies but overall structure is stable
Actors' activity/popularity changes over observation period
Focus is on individual-level temporal dynamics
Bipartite Network Models:
When mode="bipartite", the model handles rectangular adjacency matrices Y with dimensions n_A x n_B, where n_A and n_B represent the number of row and column nodes respectively.
Static Bipartite Case: The model uses separate latent factor matrices:
U: n_A x R_row matrix of row node latent positions
V: n_B x R_col matrix of column node latent positions
G: R_row x R_col interaction matrix mapping between latent spaces
Multiplicative term: U G V' captures bipartite community structure
Dynamic Bipartite Case: When dynamic_uv=TRUE for bipartite networks: $$U_{i,k,t} = \rho_{uv} U_{i,k,t-1} + \epsilon_{i,k,t}$$ $$V_{j,k,t} = \rho_{uv} V_{j,k,t-1} + \eta_{j,k,t}$$ where i indexes row nodes, j indexes column nodes, k indexes latent dimensions.
When dynamic_G=TRUE, the interaction matrix also evolves: $$G_{k,l,t} = \rho_G G_{k,l,t-1} + \xi_{k,l,t}$$ allowing the mapping between row and column latent spaces to change over time.
Key Differences from Unipartite Models:
No dyadic correlation (rho): Bipartite edges are inherently directed
Separate dimensions: R_row and R_col can differ for row/column spaces
Rectangular structure: Network density patterns differ from square matrices
Community interpretation: Captures affiliation patterns between node types
Standard AME Model Types:
The following describes the eight standard data types/models available:
"normal": A normal AME model.
"tobit": A tobit AME model for censored continuous data. Values are censored at zero, appropriate for non-negative continuous relational data.
"binary": A binary probit AME model.
"ordinal": An ordinal probit AME model. An intercept is not identifiable in this model.
"cbin": An AME model for censored binary data. The value of 'odmax' specifies the maximum number of links each row may have.
"frn": An AME model for fixed rank nomination networks. A higher value of the rank indicates a stronger relationship. The value of 'odmax' specifies the maximum number of links each row may have.
"rrl": An AME model based on the row ranks. This is appropriate if the relationships across rows are not directly comparable in terms of scale. An intercept, row random effects and row regression effects are not estimable for this model.
"poisson": An overdispersed Poisson AME model for count data. The latent variable represents the log mean of the Poisson distribution.
Examples
data(YX_bin_list)
fit<-lame(YX_bin_list$Y,YX_bin_list$X,burn=5,nscan=5,odens=1,family="binary")
# you should run the Markov chain much longer than this