GCN

GCN, or Graph Convolutional Network, is a type of neural network designed to operate on graph-structured data. In a GCN, each node carries a feature vector, and edges encode relationships. The network learns representations by repeatedly aggregating and transforming features from a node’s local neighborhood, producing node- and graph-level embeddings used for downstream tasks.

Historically, GCNs emerged from spectral graph theory and later practical spatial formulations. Spectral approaches define convolution in the graph Fourier domain, but such methods can be computationally intensive. A widely used practical variant, introduced to enable scalable training, updates node representations with a layer like H^{l+1} = σ( H^l W^l), where  is the normalized adjacency matrix with added self-loops, and W^l is a trainable weight matrix. This first-order approximation, popularized by Kipf and Welling, allows efficient learning on large graphs.

Applications of GCNs span node classification, link prediction, and graph classification. They have been applied to social networks, recommender systems, biological networks, and molecular property prediction, among other domains, leveraging their ability to propagate and integrate information from neighboring nodes.

Training considerations include dependence on an available graph structure and often semi-supervised settings when labels cover only a subset of nodes. Limitations include over-smoothing with many layers, sensitivity to graph quality and heterophily (where neighboring nodes are dissimilar), and scalability challenges for extremely large graphs. Variants and related models have broadened the field: Graph Attention Networks (GAT) introduce attention mechanisms to weight neighbors; Graph Isomorphism Networks (GIN) aim for stronger discriminative power; and numerous extensions address dynamic graphs and scalability. Overall, GCNs are a foundational approach in graph representation learning with broad adoption.