Description Usage Arguments Details Value References See Also Examples
Implements latent space models for multivariate networks (multiplex) via MCMC algorithm.
1 2 3 4 5 6 7 8 9 10 11 12 13  multiNet(Y, niter = 1000, D = 2,
muA = 0, tauA = NULL, nuA = 3,
muB = 0, tauB = NULL, nuB = 3,
muL = 0, tauL = NULL, nuL = 3,
alphaRef = NULL,
sender = c("const", "var"),
receiver = c("const", "var"),
covariates = NULL,
DIC = FALSE, WAIC = FALSE,
burnIn = round(niter*0.3),
trace = TRUE,
allChains = FALSE,
refSpace = NULL)

Y 
A threedimensional array or list of n x n adjacency matrices composing the multidimensional network. A list will be converted to an array. If an array, the dimension of 
niter 
The number of MCMC iterations. The default value is 
D 
The dimension of the latent space, with 
muA, muB, muL 
Mean hyperparameters, see details. 
tauA, tauB, tauL 
Mean hyperparameters, see details. 
nuA, nuB, nuL 
Variance hyperparameters, see details. 
alphaRef 
The value for the intercept in the first network (the reference network). This value can be specified by the user on the basis of prior knowledge. By default is computed using the function 
sender, receiver 
The type of nodespecific sender and receiver effects to be included in the model. If specified, these effects can be set to constant ( 
covariates 
An array or a list with edgecovariates matrices. A list is automatically converted to an array. Covariates can be either continuous or discrete and must be constant throughout the views of the multiplex. The dimension of 
DIC 
A logical value indicating wether the DIC (Deviance Information Criterion) should be computed. The default is 
WAIC 
A logical value indicating wether the WAIC (Widely Available Information Criterion) should be computed. The default is 
burnIn 
A numerical value, the number of iterations of the chain to be discarded when computing the posterior estimates. The default value is 
trace 
A logical value indicating if a progress bar should be printed. 
allChains 
A logical value indicating if the full parameter chains should also be returned in output. The default value is 
refSpace 
Optional. A matrix containing a set of reference values for the latent coordinates of the nodes. Its dimension must be 
The function estimates a latent space model for multidimensional networks (multiplex) via MCMC. The model assumes that the probability of observing an arc between any two nodes is inversely related to their distance in a lowdimensional latent space. Hence, nodes close in the latent space have a higher probability of being connected across the views of the multiplex than nodes far apart. The model allows the inclusion of nodespecific sender and receiver effects and edgespecific covariates.
The probability of an edge beteween nodes i and j in the k_{th} network is defined as:
P ( y_{ijk} = 1  Ω_k , d_{ij} , λ ) = C_{ijk} \ ( 1 + C_{ijk} ).
with C_{ijk} = exp( α_k  β_k * d_{ij}  λ * x_{ij} ) when nodespecific effects are not present and C_{ijk} = exp( α_k φ_{ijk}  β_k * d_{ij}  λ * x_{ij} ) when they are included in the model.
The arguments of C_{ijk} are:
The squared Euclidean distance between nodes i and j in the latent space, d_{ij}.
A coefficient λ to scale the edgespecific covariate x_{ij}. If more than one covariate is introduced in the model, their sum is considered, with each covariate being rescaled by a specific coefficient λ_l. Edgespecific covariates are assumed to be inversely related to edge probabilities, hence λ => 0 .
A vector of networkspecific parameters, Ω_k = ( α_k, β_k ). These parameters are:
A rescaling coefficient β_k, which weights the importance of the latent space in the k_{th} network, with β_k => 0. In the first network (that is the reference network), the coefficient is fixed to β_1 = 1 for identifiability reasons.
An intercept parameter α_k, which corresponds to the largest edge probability allowed in the k_{th} network. Indeed, when β_k = 0 and when no covariate is included, the probability of having a link between a couple of nodes is that of the random graph:
P ( y_{ijk} = 1  α_k ) = exp( α_k ) \ ( 1 + exp( α_k ) ).
The intercepts have a lower bound corresponding to log ( log( n ) \ ( n  log( n ) ) ). For identifiability reasons, the intercept of the first network needs to be fixed. Its value can be either specified by the user on the basis of prior knowledge or computed with the function alphaRef
.
When nodespecific effects are included in the model,
φ_{ijk} = g*(θ_{ik} + γ_{jk})
with :
θ_{ik} the sender effect of node i in network k.
γ_{k} the receiver effect of node j in network k.
g a scalar. When both sender and receiver effects are present, g=0.5; when only one type of effect is included in the model, g=1.
When the sender and/or receiver effects are set to constant ("const"
), each node i is assumed to have a constant effect across the different networks: θ_{ik} = θ_{i} and/or γ_{ik} = γ_{i}. Instead, when they are set to variable ("var"
), each node has a different effect across the networks: θ_{ik} and/or γ_{ik}.
Inference on the model parameters is carried out via a MCMC algorithm. A hierarchical framework is adopted for estimation, where the parameters of the distributions of α, β and λ are considered nuisance parameters and assumed to follow hyperprior distributions. The parameters of these hyperpriors need to be fixed and are the following:
tauA, tauB
and tauL
are the scale factors for the variances of the hyperprior distributions for the mean parameters of α_k , β_k and λ_l. If not specified by the user, tauA
and tauB
are computed as ( K  1 ) \ K , if K > 1, otherwise they are set to 0.5. Parameter tauL
is calculated as ( L  1 ) \ K , if L > 1, otherwise it is set to 0.5.
muA, muB
and muL
are the means of the hyperprior distributions for the mean parameters of α_k , β_k and λ_l. If not specified by the user, they are all set to 0.
nuA, nuB
and nuL
are the degrees of freedom of the hyperprior distributions for the variance parameters of α_k , β_k and λ_l. If not specified by the user, they are all set to 3.
Missing data are considered structural and correspond to edges missing because one or more nodes are not observable in some of the networks of the multiplex. No imputation is performed, instead, the term corresponding to the missing edge is discarded in the computation of the likelihood function. For example, if either node i or j is not observable in network k, the edge (i,j) is missing and the likelihood function for network k is calculated discarding the corresponding (i,j) term. Notice that the model assumes a single common generative latent space for the whole multidimensional network. Thus, discarding the (i,j) term in the k_{th} network does not prevent from recovering the coordinates of nodes i and j in the latent space.
An object of class 'multiNet'
containing the following components:
n 
The number of nodes in the multidimensional network. 
K 
The number of networks in the multidimensional network. 
D 
The number of dimensions of the estimated latent space. 
parameters 
A list with the following components:

latPos 
A list with posterior estimates of means and standard deviations of the latent coordinates. 
accRates 
A list with the following components:

DIC 
The Deviance Information Criterion of the estimated model. Computed only if 
WAIC 
The Widely Available Information Criterion of the estimated model. Computed only if 
allChains 
If

corrRefSpace 
A numerical vector containing the values of the Procrustes correlation between the reference space and the estimated one, computed at each mcmc iteration. Only outputed when 
info 
A list with some information on the estimated model:

D'Angelo, S. and Murphy, T. B. and Alf<c3><b2>, M. (2018). Latent space modeling of multidimensional networks with application to the exchange of votes in the Eurovision Song Contest. arXiv.
D'Angelo, S. and Alf<c3><b2>, M. and Murphy, T. B. (2018). Nodespecific effects in latent space modelling of multidimensional networks. arXiv.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  data(vickers)
it < 10 # small number of iterations just for example
# 2dimensional latent space model, no covariates
mod < multiNet(vickers, niter = it, D = 2)
# 2dimensional latent space model, sex as covariate
mod < multiNet(vickers, niter = it, D = 2,
covariates = sex)
# 2dimensional latent space model, with constant sender
# effect and variable receiver effect
mod < multiNet(vickers, niter = it, D = 2,
sender = "const", receiver = "var")

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