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Stochastic gradient descent-based inference for dynamic network models with attractors

Hancong Pan, Xiaojing Zhu, Cantay Caliskan, Dino P. Christenson, Konstantinos Spiliopoulos, Dylan Walker, Eric D. Kolaczyk

TL;DR

The paper tackles scalability in dynamic latent space networks by introducing SGD-based inference for CLSNA with attractors and a novel Laplace-inspired variance estimator. It extends CLSNA to handle nodes joining and leaving over time, enabling full-data analysis without fixed node sets. The two-stage approach achieves accurate point estimates and credible uncertainty while dramatically reducing computational time relative to MCMC, especially with GPU acceleration. Applied to the US Congress X-network, the method uncovers increasing Republican repulsion and Democratic cohesion, while mitigating selection bias inherent in fixed-sample analyses. Overall, the work provides a scalable, uncertainty-aware framework for analyzing evolving social networks with attractor dynamics.

Abstract

In Coevolving Latent Space Networks with Attractors (CLSNA) models, nodes in a latent space represent social actors, and edges indicate their dynamic interactions. Attractors are added at the latent level to capture the notion of attractive and repulsive forces between nodes, borrowing from dynamical systems theory. However, CLSNA reliance on MCMC estimation makes scaling difficult, and the requirement for nodes to be present throughout the study period limit practical applications. We address these issues by (i) introducing a Stochastic gradient descent (SGD) parameter estimation method, (ii) developing a novel approach for uncertainty quantification using SGD, and (iii) extending the model to allow nodes to join and leave over time. Simulation results show that our extensions result in little loss of accuracy compared to MCMC, but can scale to much larger networks. We apply our approach to the longitudinal social networks of members of US Congress on the social media platform X. Accounting for node dynamics overcomes selection bias in the network and uncovers uniquely and increasingly repulsive forces within the Republican Party.

Stochastic gradient descent-based inference for dynamic network models with attractors

TL;DR

The paper tackles scalability in dynamic latent space networks by introducing SGD-based inference for CLSNA with attractors and a novel Laplace-inspired variance estimator. It extends CLSNA to handle nodes joining and leaving over time, enabling full-data analysis without fixed node sets. The two-stage approach achieves accurate point estimates and credible uncertainty while dramatically reducing computational time relative to MCMC, especially with GPU acceleration. Applied to the US Congress X-network, the method uncovers increasing Republican repulsion and Democratic cohesion, while mitigating selection bias inherent in fixed-sample analyses. Overall, the work provides a scalable, uncertainty-aware framework for analyzing evolving social networks with attractor dynamics.

Abstract

In Coevolving Latent Space Networks with Attractors (CLSNA) models, nodes in a latent space represent social actors, and edges indicate their dynamic interactions. Attractors are added at the latent level to capture the notion of attractive and repulsive forces between nodes, borrowing from dynamical systems theory. However, CLSNA reliance on MCMC estimation makes scaling difficult, and the requirement for nodes to be present throughout the study period limit practical applications. We address these issues by (i) introducing a Stochastic gradient descent (SGD) parameter estimation method, (ii) developing a novel approach for uncertainty quantification using SGD, and (iii) extending the model to allow nodes to join and leave over time. Simulation results show that our extensions result in little loss of accuracy compared to MCMC, but can scale to much larger networks. We apply our approach to the longitudinal social networks of members of US Congress on the social media platform X. Accounting for node dynamics overcomes selection bias in the network and uncovers uniquely and increasingly repulsive forces within the Republican Party.
Paper Structure (23 sections, 3 theorems, 32 equations, 11 figures, 4 tables, 3 algorithms)

This paper contains 23 sections, 3 theorems, 32 equations, 11 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Extended CLSNA model
  3. Model Definition
  4. SGD-based Inference Method
  5. Point Estimate
  6. Variance Estimation
  7. The Variance Estimation Algorithm
  8. Simulation studies
  9. X Data Analysis
  10. Exploratory Analysis
  11. Results
  12. Discussion
  13. Some Useful Results of Multivariate Normal Distribution
  14. Supplementary Results in X Data Analysis
  15. BIC values
  16. ...and 8 more sections

Key Result

Theorem 7.1

Given the multivariate Gaussian distribution setting presented above, write $p(x_1, x_2)$ as the p.d.f of the joint distribution of $x_1, x_2$. $p(x_2)$ as the p.d.f of the marginal distribution of $x_2$. Then, we have $p(x_2) \propto p(x_1=\mu_{1|2},x_2)$.

Figures (11)

  • Figure 1: Numbers of re-elected and newly elected Democratic and Republican congress members in the X hashtag network
  • Figure 2: Graphical model representation of the CLSNA model. All key dependencies are shown, including edge generation, edge persistence and temporal evolution
  • Figure 3: Illustration of the Variance Estimation Method in a Bivariate Distribution
  • Figure 4: Temporal evolution of edge density within the Democratic and Republican parties, the inter-party edge density and the edge density among all members
  • Figure 5: The point estimates of the latent positions with the extended model. The dots that mainly occupy the left portion of each plot represent Democrats and the dots that mainly occupy the right portion of each plot represent Republicans. At each time point, the two parties consistently occupy different halves of the space. The Democrats exhibited clustering behavior throughout the period. Conversely, the Republicans initially displayed a similar flocking behavior, but gradually began to disperse around the year 2016.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 7.1
  • proof
  • Corollary 7.1.1
  • proof
  • Corollary 7.1.2