# Stability Properties of Graph Neural Networks

## TL;DR

This paper analyzes the stability and discriminative properties of graph neural networks, showing they are permutation equivariant and stable to topology changes, which explains their effectiveness in various applications.

## Contribution

It proves that GNNs with integral Lipschitz filters are both stable to topology changes and highly discriminative, a combination not achievable with linear filters.

## Key findings

- GNNs are permutation equivariant.
- GNNs with Lipschitz filters are stable to topology changes.
- Such GNNs effectively discriminate high-frequency information.

## Abstract

Graph neural networks (GNNs) have emerged as a powerful tool for nonlinear processing of graph signals, exhibiting success in recommender systems, power outage prediction, and motion planning, among others. GNNs consists of a cascade of layers, each of which applies a graph convolution, followed by a pointwise nonlinearity. In this work, we study the impact that changes in the underlying topology have on the output of the GNN. First, we show that GNNs are permutation equivariant, which implies that they effectively exploit internal symmetries of the underlying topology. Then, we prove that graph convolutions with integral Lipschitz filters, in combination with the frequency mixing effect of the corresponding nonlinearities, yields an architecture that is both stable to small changes in the underlying topology and discriminative of information located at high frequencies. These are two properties that cannot simultaneously hold when using only linear graph filters, which are either discriminative or stable, thus explaining the superior performance of GNNs.