- 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

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

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

Representational Capacity

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

University of Virginia, Charlottesville

GNNs are a general framework for defining deep neural networks on graph data. The goal of GNNs is to generate representations of nodes that actually depend on the structure of the graph, as well as any feature information available.

Graph Neural Networks (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

Graph Representation Learning by William Hamilton

Given an input graph $G = (V,E)$ and a set of node features **X** $\in \mathbb{R}^{d*|V|}$, our goal is to produce a node embedding $z_u$, for all nodes in the graph.

Formal Definition of GNN

One of the defining features of a GNN is that it uses a form of *neural message passing* in which vector messages are exchanged between nodes and updated using neural networks

Neural Message Passing

In the context of GNNs, many of the standard regularization approaches are known to work well, including L2 regularization, dropout, and layer normalization. However, there are also regularization strategies that are somewhat specific to the GNN setting.

Regularization in GNN

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

Original node-level message passing mechanism can be time-consuming.

Efficiency Concerns of GNNs

 $\mathcal{L} = \sum_{ \mathcal{G}_i \in \mathcal{T} } \| MLP( z_{\mathcal{G}_{i}} ) - y_{\mathcal{G}_{i}}  \|_{2}^{2}$

$z_{\mathcal{G}_{i}}$ is graph level embeddings

$\mathcal{T}  =  {\mathcal{G}_{1}, ..., \mathcal{G}_{n} }$ set of labeled training graphs

MLP ,densely connected feed forward neural network

$y_{\mathcal{G}_{i}  \in  {\rm I\!R} }$ is the target value for training graph $\mathcal{G}_{i}$


Definition of Loss in GNNs for Graph Classification

Pair wise node embedding loss function are deployed for applications utilizing graph relation prediction.

Loss function in GNNs for Relation Prediction

 - GNN pre-training approaches is an open and active area of research. 

 - Pretraining is an effective way   of injecting domain knowledge into a model before the actual training take place

 -  Pre-training a GNN using  neighborhood reconstruction loss  to reconstruct missing edges before a classification task does not improve the classification loss.

 -  A pre-training method called Deep Graph Infomax maximizes the mutual information between node embeddings $z_u$  and graph embeddings $ z_{\mathcal G}$.


Pretraining a GNN

GNNs and Graph Isomorphism

1. Given two graphs, we assign each node an initial label in each graph. 
2. We iteratively assign a new label to each node in each graph by hashing the multi-set of the current labels within the node’s neighborhood, as well as the node’s current label.
3. We repeat Step 2 until the labels for all nodes in both graphs converge.
4. Finally, we construct  multi-sets summarizing all the node labels in each graph, and we declare G1 and G2 to be isomorphic if and only if the multi-sets for both graphs are identical.

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