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Input State Stability of Gated Graph Neural Networks

Antonio Marino, Claudio Pacchierotti, Paolo Robuffo Giordano

TL;DR

This article finds the conditions for the input-to-state stability (ISS) and incremental ISS of gated graph neural networks (GGNNs) and shows that using these conditions increases the performance and robustness of the GGNNs.

Abstract

In this paper, we aim to find the conditions for input-state stability (ISS) and incremental input-state stability ($δ$ISS) of Gated Graph Neural Networks (GGNNs). We show that this recurrent version of Graph Neural Networks (GNNs) can be expressed as a dynamical distributed system and, as a consequence, can be analysed using model-based techniques to assess its stability and robustness properties. Then, the stability criteria found can be exploited as constraints during the training process to enforce the internal stability of the neural network. Two distributed control examples, flocking and multi-robot motion control, show that using these conditions increases the performance and robustness of the gated GNNs.

Input State Stability of Gated Graph Neural Networks

TL;DR

This article finds the conditions for the input-to-state stability (ISS) and incremental ISS of gated graph neural networks (GGNNs) and shows that using these conditions increases the performance and robustness of the GGNNs.

Abstract

In this paper, we aim to find the conditions for input-state stability (ISS) and incremental input-state stability (ISS) of Gated Graph Neural Networks (GGNNs). We show that this recurrent version of Graph Neural Networks (GNNs) can be expressed as a dynamical distributed system and, as a consequence, can be analysed using model-based techniques to assess its stability and robustness properties. Then, the stability criteria found can be exploited as constraints during the training process to enforce the internal stability of the neural network. Two distributed control examples, flocking and multi-robot motion control, show that using these conditions increases the performance and robustness of the gated GNNs.
Paper Structure (15 sections, 4 theorems, 38 equations, 6 figures)

This paper contains 15 sections, 4 theorems, 38 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Graph Neural Network
  4. Gated Graph Neural Network
  5. Incremental Input State Stability
  6. ONE-LAYER GGNN STABILITY
  7. DEEP GGNN STABILITY
  8. GGNN STABILITY UNDER COMMUNICATION DELAY
  9. VALIDATION EXAMPLES
  10. Flocking Control Example
  11. Multi Robot Motion Control Example
  12. CONCLUSIONS
  13. proof to Theorem \ref{['ISS_stab']}
  14. proof to Theorem \ref{['dISS_stab']}
  15. proof of Theorem \ref{['deep-delta-stability']}

Key Result

Theorem 1

A sufficient condition for the ISS of a single-layer GGNN network is that $\mathcal{A} \leq 1$, where

Figures (6)

  • Figure 1: Flocking control: a group of agents (yellow dots) move in order to reach the same velocity and to avoid collision. The leader (red dot) moves in order to reach the target (blue cross) and avoids the collision with the other agents.
  • Figure 2: Flocking and Leader Error for stable (sGGNN) and non-stable GGNN controllers, varying the team size N with fixed communication range of $4$m (with and without instantaneous communication) and the communication range with $N=25$, reported using box plots that display median, minimum, maximum, $25$th/$75$th percentiles, and outliers.
  • Figure 3: Flocking Error and Leader Error for stable and non-stable GGNN controllers with variable communication range using Laplacian
  • Figure 4: Multi Robot Motion Control: a group of agents (red dots) move to reach their targets (blue cross) avoiding agent-agent and agent-obstacle (in yellow) collisions.
  • Figure 5: Success rate and flow time for stable and non-stable GGNN controllers evaluation varying the team size, the communication range and the obstacles density for a $20m \times 20m$ map; the flow time increasing is computed as $(Tf-Tf^*)/Tf^*$ with $Tf^*$ expert controller time of arrival.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition II.1: ISS
  • Definition II.2: $\delta$ISS
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3
  • proof
  • ...and 3 more