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An Information-Theoretic Analysis of Temporal GNNs

Amirmohammad Farzaneh

TL;DR

The paper tackles the lack of theoretical tools for Temporal Graph Neural Networks by introducing an information-theoretic framework that leverages the information bottleneck and models temporal graph evolution as stochastic processes. It defines Mutual Information Rate (MIR) and a relative alternative, AMIR, establishing existence results and relations between the two, with special attention to Markov chains and Non-Equilibrium Chains (NECs) as input models. Through simulations on Gaussian processes and a two-state Hidden Markov Model, the authors demonstrate that AMIR often converges faster and can be estimated from fewer samples, while MIR provides a broader information-bound baseline. The proposed NEC-based input modeling and the AMIR/MIR toolkit offer a principled lens to assess and design temporal GNNs with improved data efficiency and theoretical grounding.

Abstract

Temporal Graph Neural Networks, a new and trending area of machine learning, suffers from a lack of formal analysis. In this paper, information theory is used as the primary tool to provide a framework for the analysis of temporal GNNs. For this reason, the concept of information bottleneck is used and adjusted to be suitable for a temporal analysis of such networks. To this end, a new definition for Mutual Information Rate is provided, and the potential use of this new metric in the analysis of temporal GNNs is studied.

An Information-Theoretic Analysis of Temporal GNNs

TL;DR

The paper tackles the lack of theoretical tools for Temporal Graph Neural Networks by introducing an information-theoretic framework that leverages the information bottleneck and models temporal graph evolution as stochastic processes. It defines Mutual Information Rate (MIR) and a relative alternative, AMIR, establishing existence results and relations between the two, with special attention to Markov chains and Non-Equilibrium Chains (NECs) as input models. Through simulations on Gaussian processes and a two-state Hidden Markov Model, the authors demonstrate that AMIR often converges faster and can be estimated from fewer samples, while MIR provides a broader information-bound baseline. The proposed NEC-based input modeling and the AMIR/MIR toolkit offer a principled lens to assess and design temporal GNNs with improved data efficiency and theoretical grounding.

Abstract

Temporal Graph Neural Networks, a new and trending area of machine learning, suffers from a lack of formal analysis. In this paper, information theory is used as the primary tool to provide a framework for the analysis of temporal GNNs. For this reason, the concept of information bottleneck is used and adjusted to be suitable for a temporal analysis of such networks. To this end, a new definition for Mutual Information Rate is provided, and the potential use of this new metric in the analysis of temporal GNNs is studied.
Paper Structure (12 sections, 3 theorems, 19 equations, 3 figures)

This paper contains 12 sections, 3 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The information bottleneck problem for temporal GNNs
  3. Alternative Mutual Information Rate (AMIR)
  4. Benefits of AMIR
  5. Rate of convergence
  6. Number of sample sequences needed
  7. AMIR of Markov chains
  8. Simulation results
  9. Gaussian processes
  10. Hidden Markov Model
  11. Application of Network Evolution Chains to temporal GNNs
  12. Conclusion

Key Result

Theorem 3

For two stationary processes $X$ and $Y$, AMIR exists.

Figures (3)

  • Figure 1: Model of an ML algorithm
  • Figure 2: Estimating MIR and AMIR
  • Figure 3: Estimating AMIR of an HMM

Theorems & Definitions (8)

  • Definition 1: Mutual Information Rate (MIR)
  • Definition 2: Alternative Mutual Information Rate (AMIR)
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • proof