To effectively construct and analyze advanced Graph Neural Network models, a solid grasp of the underlying principles is necessary. This chapter reinforces foundational concepts, viewing them through the lens required for more complex architectures and techniques.

We will start by examining methods for representing graph data for machine learning tasks, moving beyond basic adjacency matrices ($A$) to include graph Laplacians ($L = D - A$) and strategies for node feature encoding. Following this, we will take a closer look at the generalized message passing framework, often summarized as:

$$\mathbf{h}_v^{(k)} = \text{UPDATE}^{(k)} \left( \mathbf{h}_v^{(k-1)}, \text{AGGREGATE}^{(k)} \left( \{ \mathbf{h}_u^{(k-1)} : u \in \mathcal{N}(v) \} \right) \right)$$

We will analyze its theoretical properties and limitations concerning the expressive power of GNNs, particularly in relation to the Weisfeiler-Lehman (WL) graph isomorphism test.

Furthermore, we will establish the essential background in spectral graph theory needed for understanding spectral GNNs. This includes the graph Fourier transform and the concept of spectral convolutions on graphs. Concepts from graph signal processing relevant to filter design and analysis will also be introduced. By the end of this chapter, you will have solidified your understanding of these core ideas, preparing you for the advanced architectures and challenges discussed subsequently.