# Lecture 6: Stability Properties of GNNs (10/4 – 10/8)

In this lecture we analyze the stability of graph filters to additive and relative perturbations. We start defining the notion of additive perturbation discussing its meaning and consequences in graph filters. Later, we study the stability of Lipschitz filters to additive perturbations providing concrete calculations. Then, we introduce the important concept of relative perturbation with a detailed discussion of its implications and its different nature with respect to the additive perturbation model. After this, we study the stability of integral Lipschitz filters to relative perturbations.

### Video 6.1 – Additive Perturbations of Graph Filters

In this lecture we define and discuss additive perturbations, discussing their importance and limitations for the modeling of changes in graphs that can occur in some applications.

• Covers Slides 1-6 in the handout.

### Video 6.2 – Stability of Lipschitz Filters to Additive Perturbations

With the notion of additive perturbation at hand, we elaborate in this lecture about the stability of Lipschitz filters with respect to additive perturbations. We discuss with full detail the meaning of the bounds that describe the stability of Lipschitz filters.

• Covers Slides 7-14 in the handout.

### Video 6.3 – Relative Perturbations of Graph Filters

In this lecture we define and discuss the important notion of relative perturbation of graph filters. Making emphasis in the limitations of additive perturbations to model some changes that occur in graphs, we show how relative perturbations allow us to model more precisely changes that take into account the original structure of the graph.
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• Covers Slides 15-19 in the handout.

### Video 6.4 – Stability of Integral Lipschitz Filters to Relative Perturbations

In this part of the lecture we discuss the stability bounds for integral Lipschitz filters to relative perturbations. We discuss the interesting tradeoff between stability and selectivity that arises in this scenario, making emphasis in the substantial differences that find with respect to the case of additive perturbations and Lipschitz filters.

• Covers Slides 20-28 in the handout.

### Video 6.5 – Stability Properties of Graph Neural Networks

In this lecture we discuss the stability properties of graph neural networks. In particular, we show how GNNs inherit the property of stability from the graph filters and how this translate in concrete bounds similar in nature to the stability bounds obtained previously for filters.

• Covers Slides 29-38 in the handout.

### Video 6.6 – GNNs Inherit the Stability Properties of Graph Filters

In this lecture we discuss a generic inheritance proof stability where we provide deeper insights about why GNNs inherit the stability properties from filters. Part of our goal is to generalize arguments used in proofs presented before.

• Covers Slides 39-45 in the handout.