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Efficient Estimation for Longitudinal Networks via Adaptive Merging

Haoran Zhang, Junhui Wang

TL;DR

This work tackles the challenge of estimating longitudinal networks with extremely sparse temporal edges by introducing a two-step adaptive merging framework built on Poisson tensor factorization. The method first estimates on equally spaced time intervals using a low-rank Tucker representation, then adaptively merges adjacent intervals based on temporal embeddings to reduce variance while controlling bias, and finally refits under a regularized Poisson likelihood solved by a tailored projected gradient descent. The authors derive new theoretical error bounds that hold under weak and medium intensity regimes, prove consistency of the adaptive merging scheme, and demonstrate that adaptive merging yields faster convergence rates than equally spaced partitions in most scenarios. Empirical studies on synthetic data and a militarized interstate disputes dataset show substantial improvements in estimation and predictive accuracy, with interpretable adaptive time segments aligning with meaningful events.

Abstract

Longitudinal network consists of a sequence of temporal edges among multiple nodes, where the temporal edges are observed in real time. It has become ubiquitous with the rise of online social platform and e-commerce, but largely under-investigated in literature. In this paper, we propose an efficient estimation framework for longitudinal network, leveraging strengths of adaptive network merging, tensor decomposition and point process. It merges neighboring sparse networks so as to enlarge the number of observed edges and reduce estimation variance, whereas the estimation bias introduced by network merging is controlled by exploiting local temporal structures for adaptive network neighborhood. A projected gradient descent algorithm is proposed to facilitate estimation, where the upper bound of the estimation error in each iteration is established. A thorough analysis is conducted to quantify the asymptotic behavior of the proposed method, which shows that it can significantly reduce the estimation error and also provides guideline for network merging under various scenarios. We further demonstrate the advantage of the proposed method through extensive numerical experiments on synthetic datasets and a militarized interstate dispute dataset.

Efficient Estimation for Longitudinal Networks via Adaptive Merging

TL;DR

This work tackles the challenge of estimating longitudinal networks with extremely sparse temporal edges by introducing a two-step adaptive merging framework built on Poisson tensor factorization. The method first estimates on equally spaced time intervals using a low-rank Tucker representation, then adaptively merges adjacent intervals based on temporal embeddings to reduce variance while controlling bias, and finally refits under a regularized Poisson likelihood solved by a tailored projected gradient descent. The authors derive new theoretical error bounds that hold under weak and medium intensity regimes, prove consistency of the adaptive merging scheme, and demonstrate that adaptive merging yields faster convergence rates than equally spaced partitions in most scenarios. Empirical studies on synthetic data and a militarized interstate disputes dataset show substantial improvements in estimation and predictive accuracy, with interpretable adaptive time segments aligning with meaningful events.

Abstract

Longitudinal network consists of a sequence of temporal edges among multiple nodes, where the temporal edges are observed in real time. It has become ubiquitous with the rise of online social platform and e-commerce, but largely under-investigated in literature. In this paper, we propose an efficient estimation framework for longitudinal network, leveraging strengths of adaptive network merging, tensor decomposition and point process. It merges neighboring sparse networks so as to enlarge the number of observed edges and reduce estimation variance, whereas the estimation bias introduced by network merging is controlled by exploiting local temporal structures for adaptive network neighborhood. A projected gradient descent algorithm is proposed to facilitate estimation, where the upper bound of the estimation error in each iteration is established. A thorough analysis is conducted to quantify the asymptotic behavior of the proposed method, which shows that it can significantly reduce the estimation error and also provides guideline for network merging under various scenarios. We further demonstrate the advantage of the proposed method through extensive numerical experiments on synthetic datasets and a militarized interstate dispute dataset.
Paper Structure (14 sections, 7 theorems, 28 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 7 theorems, 28 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Poisson point process and tensor factorization
  4. Adaptive merging
  5. Regularized likelihood estimation
  6. Computation
  7. Theory
  8. A new error bound for the PGD algorithm
  9. Error analysis based on equally spaced intervals
  10. Error analysis based on adaptively merged intervals
  11. Numerical experiments
  12. Simulation examples
  13. Real example
  14. Discussion

Key Result

Theorem 1

Let $\overline\mathcal{M} = [\overline\mathcal{S};\overline\mathbf U,\overline\mathbf V,\overline\mathbf W]\in\mathbb R^{n_1\times n_2\times n_3}$ be a pre-specified order-3 tensor with $\overline\mathcal{S}\in\mathcal{C}_{\mathcal{S}},~\overline\mathbf U\in\mathcal{C}_{\mathbf U}$, $\overline\mathb for some constant $0<\kappa<1$ and $c_1>0$.

Figures (4)

  • Figure 1: Flowchart for the estimation procedure, where $n = \max\{n_1,n_2\}$ and the logarithmic factors are suppressed.
  • Figure 2: The average tensor estimation errors based on equal spaced intervals under three scenarios of Table \ref{['tab:res']} with different values of $L$ over 50 independent replications. The red dotted lines are the average estimation errors of the estimate based on adaptively merged intervals. The large error rates in the third panel is due to the much smaller chosen $T$ in the third scenario.
  • Figure 3: The estimated temporal embedding vectors $\{\widehat{\mathbf w}_{l,\boldsymbol \delta}\}_{l=1}^9$, where colors represent different merged time intervals.
  • Figure 4: The average estimation errors of AM($\widehat{K}$) and ES($L$) in each of the $L=24$ intervals over 50 replications.

Theorems & Definitions (13)

  • Remark 1
  • Theorem 1
  • Proposition 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Theorem 3
  • Remark 5
  • ...and 3 more