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Mean-Field Analysis of Latent Variable Process Models on Dynamically Evolving Graphs with Feedback Effects

Ankan Ganguly, Konstantinos Spiliopoulos, Daniel Sussman

TL;DR

The paper develops a rigorous mean-field framework for dynamic co-evolving latent-space networks with persistence, establishing a limiting description for a randomly sampled subnetwork and proving a rich conditional propagation of chaos. It introduces a comprehensive limiting model for latent states and edges, derives hydrodynamic limits, and shows graphon and multigraphon convergence for the evolving network trajectories. The results hinge on a careful sample-based perspective, conditional independence structures, and multiplex graph representations, complemented by numerical verification. The methodology provides a principled way to understand asymptotic behavior in large, endogenous networks with feedback between states and topology, with implications for analyzing opinion dynamics and other social processes on dynamic networks.

Abstract

In this paper, we study the asymptotic behavior of a class of dynamic co-evolving latent space networks. The model we study is subject to bi-directional feedback effects, meaning that at any given time, the latent process depends on its own value and the graph structure at the previous time step, and the graph structure at the current time depends on the value of the latent processes at the current time but also on the graph structure at the previous time instance (sometimes called a persistence effect). We construct the mean-field limit of this model, which we use to characterize the limiting behavior of a random sample taken from the latent space network in the limit as the number of nodes in the network diverges. From this limiting model, we can derive the limiting behavior of the empirical measure of the latent process and establish the related graphon limit of the latent particle network process. We also provide a description of the rich conditional probabilistic structure of the limiting model. The inherent dependence structure complicates the mathematical analysis significantly. In the process of proving our main results, we derive a general conditional propagation of chaos result, which is of independent interest. In addition, our novel approach of studying the limiting behavior of random samples proves to be a very useful methodology for fully grasping the asymptotic behavior of co-evolving particle systems. Numerical results are included to illustrate the theoretical findings.

Mean-Field Analysis of Latent Variable Process Models on Dynamically Evolving Graphs with Feedback Effects

TL;DR

The paper develops a rigorous mean-field framework for dynamic co-evolving latent-space networks with persistence, establishing a limiting description for a randomly sampled subnetwork and proving a rich conditional propagation of chaos. It introduces a comprehensive limiting model for latent states and edges, derives hydrodynamic limits, and shows graphon and multigraphon convergence for the evolving network trajectories. The results hinge on a careful sample-based perspective, conditional independence structures, and multiplex graph representations, complemented by numerical verification. The methodology provides a principled way to understand asymptotic behavior in large, endogenous networks with feedback between states and topology, with implications for analyzing opinion dynamics and other social processes on dynamic networks.

Abstract

In this paper, we study the asymptotic behavior of a class of dynamic co-evolving latent space networks. The model we study is subject to bi-directional feedback effects, meaning that at any given time, the latent process depends on its own value and the graph structure at the previous time step, and the graph structure at the current time depends on the value of the latent processes at the current time but also on the graph structure at the previous time instance (sometimes called a persistence effect). We construct the mean-field limit of this model, which we use to characterize the limiting behavior of a random sample taken from the latent space network in the limit as the number of nodes in the network diverges. From this limiting model, we can derive the limiting behavior of the empirical measure of the latent process and establish the related graphon limit of the latent particle network process. We also provide a description of the rich conditional probabilistic structure of the limiting model. The inherent dependence structure complicates the mathematical analysis significantly. In the process of proving our main results, we derive a general conditional propagation of chaos result, which is of independent interest. In addition, our novel approach of studying the limiting behavior of random samples proves to be a very useful methodology for fully grasping the asymptotic behavior of co-evolving particle systems. Numerical results are included to illustrate the theoretical findings.

Paper Structure

This paper contains 45 sections, 39 theorems, 222 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Assumptions and Limiting Behavior of the Latent Particle System
  4. Assumptions
  5. Limiting Model over a Randomly Sampled Subnetwork
  6. Building Intuition
  7. A Full Characterization of the Limiting Model
  8. Convergence Results
  9. Graphon and Multigraphon Convergence of the Latent Particle Network Process
  10. Left-Convergence of Graphs and Graphon Convergence
  11. Convergence of Latent Network Trajectories as Probability Graphons
  12. The Latent Network Trajectory Limit
  13. Numerical Results
  14. Algorithms
  15. A Mean-Field Verification Algorithm
  16. ...and 30 more sections

Key Result

Proposition 2.6

Definitions defn::AZlim(a), defn::AZlim(b), and defn::AZlim(c) individually characterize the distribution of $(Z^{(1:k)}[t],A^{(1:k,1:k)}[t])$. In addition, the distributions described in all three parts of Definition defn::AZlim are equivalent.

Figures (4)

  • Figure 1: The Mean Square Error of the mean-field approximation of the particle trajectories averaged over all particles and simulations.
  • Figure 2: The density of edges in the symmetric difference network $G^{n,k}_{SD}(t)$.
  • Figure 3: In Figures \ref{['fig::trianglePlot']}(a)-(c), we examine the average error we would find if we tried to approximate the homomorphism densities of triangles in the $n$-particle system using the limiting mean-field process. In Figure (d), we compare the homomorphism density of triangles in the mean-field model vs. an Erdös-Rènyi graph with the same edge density as the mean-field model.
  • Figure 4: The average difference in the 2nd largest eigenvalue of the mean-field model and the $n$ particle model (divided by $n$).

Theorems & Definitions (97)

  • Example 1.1: BCOD Model
  • Example 1.2: CLSNA Model
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6: Conditional Structure of the Limiting Model
  • Corollary 2.7
  • Theorem 2.8: Asymptotic Distribution of a Random Sample
  • Theorem 2.9: Hydrodynamic Limits
  • Remark 2.10
  • Proposition 3.1
  • ...and 87 more