1.  The WL algorithm aggregates and updates discrete labels (using a hash function) while GNN models aggregate and update node embeddings using neural networks.
2. GNNs have been motivated and derived as a continuous and differentiable analog of the WL algorithm.


The key distinction between GNN and the WL algorhtm

The relationship between GNNs and the WL algorithm

1. The key distinction between GNN and WL.
2. The relationship between GNNs and the WL algorithm.

University of Illinois at Urbana-Champaign

The basic GNN can be derived as a neural network variation of the existing Weisfieler-Lehman (WL) isomorphism algorithm

GNNs and Graph Isomorphism

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

Graph Representation Learning by William Hamilton

- are essentially identical
- have the same same graph structure

The only differences that can be noted in Isomorphic graphs are that they can have a different ordering of the nodes in their adjacency matrix. This is because the ordering has no bearing on the structure of the graph, it is just an arbitrary decision in representing a graph using algebraic objects

Characteristics of Isomorphic Graphs:

Goal of Isomorphism Testing: declare whether or not two graphs are isomorphic

- Testing for graph isomorphism is fundamentally a hard problem
- Referred to as NP-indeterminate, no polynomial-time algorithms are known for the problem

Testing for Graph Isomorphism

GNNs and the WL Algorithm

Graph isomorphism testing gives us a way to quantify the representational power of different learning approaches

Given 2 learned representations z_g1 and z_g2 for two corresponding graphs:
- a "perfect" algorithm would have z_g1 = z_g2 if and only if g1 is isomorphic to g2

However, there is no "perfect" representation learning algorithm, but there are several ways to connect it to graph isomorphism testing and approximate its power, such as the The Weisfieler-Lehman (WL) Algorithm

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