Input State Stability of Gated Graph Neural Networks
In this paper, we aim to find the conditions for input-state stability (ISS)
and incremental input-state stability ($\delta$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.
Antonio Marino, Claudio Pacchierotti, Paolo Robuffo Giordano
# On Stability of Gated Graph Neural Networks

###### Abstract

In this paper, we aim to find the conditions for input-state stability (ISS) and incremental input-state stability (\(\delta\)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.

 distributed control, graph neural network, stability analysis

## I Introduction

Multi-agent systems have been successfully studied in the past few years [1]. With respect to single-agent approaches, coordinated multi-agent systems are expected to collaboratively solve tasks and offer more flexibility, all features that make these systems suited to solve problems in a variety of disciplines including computer science, electrical engineering, and robotics [2]. The collaborative control of multiple agents must take into account the needs of the group. For multi-robot applications, individual robot motion should be generated by using not only local sensing data, but also knowledge about the group state, usually retrieved through communication with a limited number of (neighboring) team members [3]. Hence, communication is one of the key elements to realize distributed solution in multi-agent systems.

In the last decade, the control community has widely adopted neural networks in data-driven control applications, taking advantage of their superior approximation capabilities [4]. In distributed control, neural networks are particularly convenient since they can approximate distributed policies without the need of cumbersome optimizations and designs. In the literature, we can find multiple examples of data driven approaches especially based on reinforcement learning [5, 6] fed by input data such as images [7]. However, these approaches work on local sensing data without communicating with the other agents. The communication is used in distributed machine learning where the learning process is partitioned on several machines that contribute to the group knowledge [8]. On the contrary, in data-driven distributed control, a recent trend involves using Graph Neural Networks (GNNs) to encode distributed control and planing solutions.

GNNs perform prediction and analysis on graphs, a convenient topological representation in different kinds of problems like text classification, protein interface predictions, and social network decisions [9]. For these latter, GNNs are more effective than the classical neural network architectures [10]. At the same time, they gave a new perspective in realization of distributed control.

Gama et al. [11] use GNNs to define a distributed LQR controller, casting the linear-quadratic problem as a self-supervised learning to find the best GNN-based distributed control, thanks to the native GNN distributed nature. The same authors [12] develop a GNN-based flocking control deployed on large team scale. Further examples can be found in space coverage [13], multi robots path [14] and motion planning [15] also in obstacles rich environment GNN [16]. GNNs are used also in approaches for enhancing multi agents perception [17] or performing a distributed active information acquisition [18] translating the multi-robot information gathering problem to a graph representation and formulate GNN-based decision maker.

In the recent literature, a very relevant discussion is about making the data-driven based method robust and stable [19]. In this context, many works applied contraction analysis to demonstrate recurrent neural network stability [20], or directly closed-loop stability in continuous learning [21] and adaptive control [22]. Other works have attempted to formulate new neural network models for achieving closed-loop stability, such as [23] where the controller is obtained from an Hamiltonian Deep Neural Networks and the stability is guaranteed by the compositional properties of port-Hamiltonian systems. Recently, the authors in [24, 25] demonstrated the ISS and incremental ISS (\(\delta\)ISS) [26] for LSTMs and GRUs, two of the most popular recurrent neural network models.

Inspired by these last results, the goal of this work is to characterize the \(\delta\)ISS properties of the recurrent version of GNN, i.e. Gated Graph Neural Networks (GGNN) [27]. These models use gated mechanisms to deploy distributed recurrent models able to reason on temporal- and spatial-based relationships among the agents. The \(\delta\)ISS property is a stronger property than plain ISS and leads to asymptotically convergence of two state trajectories when their respective input sequences are close, regardless of the initial conditions for the states. Therefore, we directly focus on the incremental stability results avoiding the complexity of other frameworks, e.g., contraction analysis.

To the best of our knowledge, this is the first time ISS and \(\delta\)ISS are proven for GGNN, whereas previous works have focused on limited stability results like stability to graph perturbations [27, 28]. Instead, this letter considers