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A Bayesian Dynamic Latent Space Model for Weighted Networks

Roberto Casarin, Matteo Iacopini, Antonio Peruzzi

Abstract

A new dynamic latent space eigenmodel (LSM) is proposed for weighted temporal networks. The model accommodates integer-valued weights, excess of zeros, time-varying node positions (features), and time-varying network sparsity. The latent positions evolve according to a vector autoregressive process that accounts for lagged and contemporaneous dependence across nodes and features, a characteristic neglected in the LSM literature. A Bayesian approach is used to address two of the primary sources of inference intractability in dynamic LSMs: latent feature estimation and the choice of latent space dimension. We employ an efficient auxiliary-mixture sampler that performs data augmentation and supports conditionally conjugate prior distributions. A point-process representation of the network weights and the finite-dimensional distribution of the latent processes are used to derive a multi-move sampler in which each feature trajectory is drawn in a single block, without recursions. This sampling strategy is new to the network literature and can significantly reduce computational time while improving chain mixing. To avoid trans-dimensional samplers, a Laplace approximation of the partial marginal likelihood is used to design a partially collapsed Gibbs sampler. Overall, our procedure is general, as it can be easily adapted to static and dynamic settings, as well as to other discrete or continuous weight distributions.

A Bayesian Dynamic Latent Space Model for Weighted Networks

Abstract

A new dynamic latent space eigenmodel (LSM) is proposed for weighted temporal networks. The model accommodates integer-valued weights, excess of zeros, time-varying node positions (features), and time-varying network sparsity. The latent positions evolve according to a vector autoregressive process that accounts for lagged and contemporaneous dependence across nodes and features, a characteristic neglected in the LSM literature. A Bayesian approach is used to address two of the primary sources of inference intractability in dynamic LSMs: latent feature estimation and the choice of latent space dimension. We employ an efficient auxiliary-mixture sampler that performs data augmentation and supports conditionally conjugate prior distributions. A point-process representation of the network weights and the finite-dimensional distribution of the latent processes are used to derive a multi-move sampler in which each feature trajectory is drawn in a single block, without recursions. This sampling strategy is new to the network literature and can significantly reduce computational time while improving chain mixing. To avoid trans-dimensional samplers, a Laplace approximation of the partial marginal likelihood is used to design a partially collapsed Gibbs sampler. Overall, our procedure is general, as it can be easily adapted to static and dynamic settings, as well as to other discrete or continuous weight distributions.
Paper Structure (16 sections, 3 theorems, 20 equations, 4 figures)

This paper contains 16 sections, 3 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Scope and Contribution
  3. Related literature
  4. Structure and notation
  5. A Dynamic Zero-Inflated Counts Eigenmodel
  6. Model
  7. Data Augmentation
  8. Parametrisation
  9. Bayesian Inference
  10. Parameter prior specification
  11. Posterior sampling
  12. Real-data Applications
  13. UN Co-voting Networks
  14. Trade Networks
  15. Brain Networks
  16. ...and 1 more sections

Key Result

Lemma 1

Define $\boldsymbol{\tau}_{ij,t} = (\tau_{ij, 1t}, \tau_{ij, 2t})$ and $\mathbf{r}_{ij,t} = (r_{ij, 1t},r_{ij, 2t})$ and assume $w_{ij,t}=1$, then the Poisson distribution in eq. eq:model_yt is the marginal distribution of $p(y_{ij,t}, \boldsymbol{\tau}_{ij,t}, \mathbf{r}_{ij,t} \mid w_{ij,t}=1, \m where $g_1(\tau_{ij,1t})$ and $g_2(\tau_{ij,2t})$ are the densities of $\mathcal{N}( \! -\log(\tau_

Figures (4)

  • Figure 1: UN General Assembly co-voting network. Circular projection of the latent space representation for the years 2014 and 2024 (left and middle), and node correlation posterior mean (right).
  • Figure 2: Trade network. Latent space representation in the $(\mathbf{x}_{:1,t},\boldsymbol{\alpha})$-plane (left) in 2024 with 95% credible ellipses which are reported in blue, and position temporal evolution (right), with lighter node colours denoting more recent years.
  • Figure 3: Spatial and distributional characteristics of the parameter estimates. Top: Brain regions (Schaefer-200 atlas) coloured by node-specific posterior mean $\hat{\alpha}_i$. Bottom left: Posterior distribution of the average effects over left ($\sum_{i=1}^{100}\alpha_i/100$, light gray) and right hemisphere nodes ($\sum_{i=101}^{200}\alpha_i/100$, dark gray). Bottom right: $\hat{\Sigma}$, estimated covariance matrix among $\mathbf{x}_{:\ell, t}$, with dashed lines separating hemispheres.
  • Figure 4: Results across five MRI sessions (rows). (a) Latent space representations. Estimated coordinate pairs $(\hat{x}_{i,1,t},\hat{x}_{i,2,t})$ and $(\hat{x}_{i,5,t},\hat{x}_{i,6,t})$ (left and middle columns) for a subsample $t=2,4$. Each point is a brain region as defined by Schaefer's 200 atlas: left hemisphere (orange) and right hemisphere (blue). (b) Posterior probability of structural zeroes, $\mathbb{P}(z_{ij,t}\leq 0\mid\mathbf{y})$. A darker colour denotes a higher probability of a structural zero.

Theorems & Definitions (4)

  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Remark