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Embedding Network Autoregression for time series analysis and causal peer effect inference

Jae Ho Chang, Subhadeep Paul

TL;DR

The paper develops Embedding Network Autoregression (ENAR) and its additive-multiplicative extension (AMNAR) to jointly handle multivariate networked time series and causal peer influence with latent homophily. By embedding latent positions from Random Dot Product Graphs or latent-space models into the NAR framework, the authors derive consistency and asymptotic normality results under growing network size $N$, time horizon $T$, and latent-dimension $K$, with distinct rates for latent and outcome parameters. They propose a model-selection criterion to choose the latent dimension and study biases arising from omitting latent vectors, providing guidance on when eigenvectors or spectral embeddings should be used. Simulations and real-data examples (Knecht dataset and wind-speed networks) demonstrate robust performance, improved prediction, and reliable inference under latent confounding and model misspecification. The contributions offer a principled, scalable approach for structure-aware time-series prediction and causal inference in networks, with practical latency-aware dimension selection and theoretical guarantees.

Abstract

We propose an Embedding Network Autoregressive Model for multivariate networked longitudinal data. We assume the network is generated from a latent variable model, and these unobserved variables are included in a structural peer effect model or a time series network autoregressive model as additive effects. This approach takes a unified view of two related yet fundamentally different problems: (1) modeling and predicting multivariate networked time series data and (2) causal peer influence estimation in the presence of homophily from finite time longitudinal data. Our estimation strategy comprises estimating latent variables from the observed network followed by least squares estimation of the network autoregressive model. We show that the estimated momentum and peer effect parameters are consistent and asymptotically normally distributed in setups with a growing number of network vertices (N) while considering both a growing number of time points T (for the time series problem) and finite T cases (for the peer effect problem). We allow the number of latent vectors K to grow at appropriate rates, which improves upon existing rates when such results are available for related models. Our theoretical results encompass cases both when the network is modeled with the random dot product graph model (ENAR) and a more general latent space model with both additive and multiplicative effects (AMNAR). We also develop a selection criterion when K is unknown that provably does not under-select and show that the theoretical guarantees hold with the selected number for K as well. Interestingly, even though we propose a unified model, our theoretical results find that different growth rates and restrictions on the latent vectors are needed to induce omitted variable bias in the peer effect problem and to ensure consistent estimation in the time series problem.

Embedding Network Autoregression for time series analysis and causal peer effect inference

TL;DR

The paper develops Embedding Network Autoregression (ENAR) and its additive-multiplicative extension (AMNAR) to jointly handle multivariate networked time series and causal peer influence with latent homophily. By embedding latent positions from Random Dot Product Graphs or latent-space models into the NAR framework, the authors derive consistency and asymptotic normality results under growing network size , time horizon , and latent-dimension , with distinct rates for latent and outcome parameters. They propose a model-selection criterion to choose the latent dimension and study biases arising from omitting latent vectors, providing guidance on when eigenvectors or spectral embeddings should be used. Simulations and real-data examples (Knecht dataset and wind-speed networks) demonstrate robust performance, improved prediction, and reliable inference under latent confounding and model misspecification. The contributions offer a principled, scalable approach for structure-aware time-series prediction and causal inference in networks, with practical latency-aware dimension selection and theoretical guarantees.

Abstract

We propose an Embedding Network Autoregressive Model for multivariate networked longitudinal data. We assume the network is generated from a latent variable model, and these unobserved variables are included in a structural peer effect model or a time series network autoregressive model as additive effects. This approach takes a unified view of two related yet fundamentally different problems: (1) modeling and predicting multivariate networked time series data and (2) causal peer influence estimation in the presence of homophily from finite time longitudinal data. Our estimation strategy comprises estimating latent variables from the observed network followed by least squares estimation of the network autoregressive model. We show that the estimated momentum and peer effect parameters are consistent and asymptotically normally distributed in setups with a growing number of network vertices (N) while considering both a growing number of time points T (for the time series problem) and finite T cases (for the peer effect problem). We allow the number of latent vectors K to grow at appropriate rates, which improves upon existing rates when such results are available for related models. Our theoretical results encompass cases both when the network is modeled with the random dot product graph model (ENAR) and a more general latent space model with both additive and multiplicative effects (AMNAR). We also develop a selection criterion when K is unknown that provably does not under-select and show that the theoretical guarantees hold with the selected number for K as well. Interestingly, even though we propose a unified model, our theoretical results find that different growth rates and restrictions on the latent vectors are needed to induce omitted variable bias in the peer effect problem and to ensure consistent estimation in the time series problem.
Paper Structure (64 sections, 35 theorems, 225 equations, 17 figures, 4 tables)

This paper contains 64 sections, 35 theorems, 225 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Connection to related literature
  3. Main results and contributions
  4. Network time series results:
  5. Peer effect inference results
  6. Selection of K and results with model misspecification
  7. Omitted variable bias when latent vectors are ignored
  8. Difference between eigenvectors and spectral embedding vectors
  9. Latent space model (LSN) results
  10. Embedding Network Autoregressive Model
  11. Network Vector Autoregression
  12. Random Dot Product Graph
  13. Embedding Network Autoregression
  14. Finite time model for peer influence
  15. Additive and multiplicative latent variables from latent space
  16. ...and 49 more sections

Key Result

Theorem 3.1

If $|\alpha|+|\theta|<1$, then there is a unique strictly stationary solution to the ENAR model eq-enar_vec with a finite first moment, and the solution is given by, where the latent variable $\mathbf{U}$ is either $\mathbf{U}_{ev}$ or $\mathbf{U}_{se}$.

Figures (17)

  • Figure 1: Causal diagram for peer effects
  • Figure 2: Boxplot of estimates of $\theta$ (left) and $\alpha$ (right) from ENAR and NAR model with increasing $N$ when data is generated from ENAR model with DCMMSBM. The rows corresponds to $K=3$ and $K=12$, and the columns corresponds to $T=40, 160, 320$.
  • Figure 3: Boxplot of the prediction error from ENAR, CNAR, and NAR model when data is generated from ENAR model with DCMMSBM and DCSBM respectively.
  • Figure 4: Model misspecification: Boxplot of prediction error from CNAR, ENAR, and NAR model when data is generated from CNAR and NAR models respectively.
  • Figure 5: Boxplot of estimates of $\theta$ and $\alpha$ from ENAR, and NAR model when data is generated from ENAR and NAR model with DCMMSBM.
  • ...and 12 more figures

Theorems & Definitions (103)

  • Definition 1
  • Theorem 3.1
  • Lemma 3.1
  • Theorem 3.2
  • Definition 2: Xie2019EfficientEF
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • ...and 93 more