Given a discrete time-varying signal $f(t_0),f(t_2),...,f(t_{N−1})$, we can represent this signal as a chain (or cycle) graph. Each node in the graph represents a point in time $t$ and the signal value at that node represents the function value $f(t)$

Representing discrete time-varying signals with a graph

In order to generalize the notion of a convolution to graphs, we first must define
what we wish to generalize and provide some brief background details. We can define the general continuous convolution operation * shown below, let f and h be two functions.


 The General Continuous Convolution Operation

There’re several types of GNN inspired by convolutions operations and their conenctions:

1)	GNN That Use Purely Convolutional Approaches

2)	Graph Convolutional Networks (GCN)

3)	GNN Without Message Passing


Convolution-Inspired GNNs

 In this perspective, where GNNs were developed as a generalization of Euclidean convolutions to the non-Euclidean graph domain, they first arise from the question: How can we generalize the notion of convolutions to general graph-structured data?

University of Michigan - Ann Arbor

GNNs had been independently developed from several distinct theoretical motivations and perspectives:
- Graph convolutions: the theory of graph signal processing, as a generalization of Euclidean convolutions to the non-Euclidean graph domain.
- Probabilistic graphical models: neural message passing approaches proposed by analogy to message passing algorithms for probabilistic inference in graphical models.
- Graph isomorphism: connection to the Weisfeiler-Lehman graph isomorphism
test.

Theoretical Motivations of GNNs

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

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