The fully-supervised node classification loss is typically the negative log-likelihood of the true label under a softmax classifier applied to each node's embedding: $$\mathcal{L} = -\sum_{v_i \in \mathcal{V}_{train}} \log\big(\text{softmax}(\text{MLP}(z_i))_{y_i}\big)$$ where $$z_i$$ is the GNN-produced embedding for node $$v_i$$, $$\mathcal{V}_{train}$$ is the set of labeled training nodes, and $$y_i$$ is the true label of node $$v_i$$.

Node Classification Loss Formula in GNNs

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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University of Michigan - Ann Arbor

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

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.

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.

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