On the Nonasymptotic Scaling Guarantee of Hyperparameter Estimation in Inhomogeneous, Weakly-Dependent Complex Network Dynamical Systems

2026-01-22

AI Summary

This work addresses the lack of non-asymptotic guarantees for hierarchical Bayesian hyperparameter estimation in large-scale, heterogeneous, and weakly dependent complex networks—a gap that risks estimator divergence as network size grows. From a measure transport perspective, the paper develops a hyperparameter estimation framework tailored to mean-field observations and establishes non-asymptotic error bounds for various optimization algorithms under fixed observation horizons. It provides the first non-asymptotic bias bound that explicitly characterizes the dependence of hyperparameter estimation on network size in heterogeneous systems, extending consistency results from independent nodes to weakly dependent settings. Experiments on SIS epidemic models and spiking neural networks confirm that estimation error diminishes with increasing network size, aligning with theoretical predictions and demonstrating the statistical consistency of the proposed approach.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Hierarchical Bayesian models are increasingly used in large, inhomogeneous complex network dynamical systems by modeling parameters as draws from a hyperparameter-governed distribution. However, theoretical guarantees for these estimates as the system size grows have been lacking. A critical concern is that hyperparameter estimation may diverge for larger networks, undermining the model's reliability. Formulating the system's evolution in a measure transport perspective, we propose a theoretical framework for estimating hyperparameters with mean-type observations, which are prevalent in many scientific applications. Our primary contribution is a nonasymptotic bound for the deviation of estimate of hyperparameters in inhomogeneous complex network dynamical systems with respect to network population size, which is established for a general family of optimization algorithms within a fixed observation duration. While we firstly establish a consistency result for systems with independent nodes, our main result extends this guarantee to the more challenging and realistic setting of weakly-dependent nodes. We validate our theoretical findings with numerical experiments on two representative models: a Susceptible-Infected-Susceptible model and a Spiking Neuronal Network model. In both cases, the results confirm that the estimation error decreases as the network population size increases, aligning with our theoretical guarantees. This research proposes the foundational theory to ensure that hierarchical Bayesian methods are statistically consistent for large-scale inhomogeneous systems, filling a gap in this area of theoretical research and justifying their application in practice.

Problem

• hyperparameter estimation
• complex network dynamical systems
• nonasymptotic guarantee
• weak dependence
• inhomogeneous systems

Innovation

• nonasymptotic guarantee
• hyperparameter estimation
• weak dependence
• measure transport
• hierarchical Bayesian models