The loss function for Graph Neural Networks (GNNs) in graph classification tasks is defined as:

$$\mathcal{L} = \sum_{\mathcal{G}_i \in \mathcal{T}} \| \text{MLP}(z_{\mathcal{G}_i}) - y_{\mathcal{G}_i} \|_2^2$$

Where:
- $$z_{\mathcal{G}_i}$$ represents the graph-level embeddings.
- $$\mathcal{T} = \{\mathcal{G}_1, \dots, \mathcal{G}_n\}$$ is the set of labeled training graphs.
- $$\text{MLP}$$ is a densely connected feed-forward neural network.
- $$y_{\mathcal{G}_i} \in \mathbb{R}$$ is the target value for the training graph $$\mathcal{G}_i$$.

Graph Classification Loss Formula in GNNs

2. Graph classification: Similar to node classification, applications on graph-level classification are popular as benchmark tasks. Historically, kernel methods were popular for graph classification, and—as a result—some of the most popular early benchmarks for graph classification were adapted from the kernel literature, such as tasks involving the classification of enzyme properties based on graph-based representations.

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For most current applications, graph neural networks are used in one of three tasks: node classification, graph classification, or relation prediction.

Applications and Loss Functions

Graph neural networks graph classification

Graph neural networks (GNNs) are commonly applied to relation prediction tasks, such as recommender systems and knowledge graph completion. In these contexts, pre-training GNNs with a neighborhood reconstruction loss can be a useful strategy to improve performance on downstream classification tasks.

Relation Prediction with Graph Neural Networks

Node classification is one of the most common benchmark tasks for graph neural networks (GNNs). The standard approach trains a GNN in a fully-supervised manner: node embeddings are passed through a classifier (e.g., an MLP followed by a softmax) and the model is optimized by minimizing the negative log-likelihood (cross-entropy) loss between predicted and true node labels.

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