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High Precision Variational Bayesian Inference of Sparse Linear Networks

Junyang Jin, Ye Yuan, Jorge Goncalves

TL;DR

The paper tackles the challenge of inferring sparse, partially observed networks with high precision. It develops a variational Bayesian framework on dynamical structure functions, using Gaussian-process priors with Tuned/Correlated kernels and backward link selection to enforce sparsity while promoting stability. Across extensive simulations, the method achieves 100% or near-100% precision and outperforms kernel-based empirical Bayes approaches under varying topology, data length, and noise, demonstrating robustness to hidden states. This approach enables reliable topology identification in domains like biology and power systems without requiring full-state measurements, with potential extensions to nonlinearity and alternative kernels.

Abstract

Sparse networks can be found in a wide range of applications, such as biological and communication networks. Inference of such networks from data has been receiving considerable attention lately, mainly driven by the need to understand and control internal working mechanisms. However, while most available methods have been successful at predicting many correct links, they also tend to infer many incorrect links. Precision is the ratio between the number of correctly inferred links and all inferred links, and should ideally be close to 100%. For example, 50% precision means that half of inferred links are incorrect, and there is only a 50% chance of picking a correct one. In contrast, this paper develops a method, based on variational Bayesian inference and Gaussian processes, that focuses on inferring links with very high precision. In addition, our method does not require full-state measurements and effectively promotes both system stability and network sparsity. Monte Carlo simulations illustrate that our method has 100% or nearly 100% precision, even in the presence of noise. The method should be applicable to a wide range of network inference contexts, including biological networks and power systems.

High Precision Variational Bayesian Inference of Sparse Linear Networks

TL;DR

The paper tackles the challenge of inferring sparse, partially observed networks with high precision. It develops a variational Bayesian framework on dynamical structure functions, using Gaussian-process priors with Tuned/Correlated kernels and backward link selection to enforce sparsity while promoting stability. Across extensive simulations, the method achieves 100% or near-100% precision and outperforms kernel-based empirical Bayes approaches under varying topology, data length, and noise, demonstrating robustness to hidden states. This approach enables reliable topology identification in domains like biology and power systems without requiring full-state measurements, with potential extensions to nonlinearity and alternative kernels.

Abstract

Sparse networks can be found in a wide range of applications, such as biological and communication networks. Inference of such networks from data has been receiving considerable attention lately, mainly driven by the need to understand and control internal working mechanisms. However, while most available methods have been successful at predicting many correct links, they also tend to infer many incorrect links. Precision is the ratio between the number of correctly inferred links and all inferred links, and should ideally be close to 100%. For example, 50% precision means that half of inferred links are incorrect, and there is only a 50% chance of picking a correct one. In contrast, this paper develops a method, based on variational Bayesian inference and Gaussian processes, that focuses on inferring links with very high precision. In addition, our method does not require full-state measurements and effectively promotes both system stability and network sparsity. Monte Carlo simulations illustrate that our method has 100% or nearly 100% precision, even in the presence of noise. The method should be applicable to a wide range of network inference contexts, including biological networks and power systems.

Paper Structure

This paper contains 19 sections, 44 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. OVERVIEW OF VARIATIONAL INFERENCE
  3. MODEL FORMULATION
  4. The dynamical structure function
  5. The likelihood distribution
  6. The prior distributions
  7. The full Bayesian model
  8. Estimation of hyperparameters
  9. VARIATIONAL INFERENCE OF DYNAMICAL STRUCTURE FUNCTIONS
  10. Update of random quantities
  11. Lower bound of model evidence
  12. Algorithm for variational inference
  13. Algorithm implementation
  14. Detection of network topology
  15. Theoretical analysis of the algorithm
  16. ...and 4 more sections

Figures (4)

  • Figure 1: The structure of a randomly generated network. Solid lines with arrows represent links. Red circles denote nodes.
  • Figure 2: Prediction of randomly generated networks.
  • Figure 3: A network with the ring structure. Symbol '$\sim$' denotes the input signals.
  • Figure 4: Prediction of ring networks.