GCNs

Graph convolutional networks (GCNs) are neural networks designed to operate on graph-structured data. They extend the convolution operation to graphs by learning node representations through iterative neighborhood aggregation. In a typical GCN layer, each node updates its feature vector by combining the features of its neighbors, weighted by a normalized adjacency structure. A common formulation uses the normalized symmetric adjacency matrix: H^{l+1} = sigma( D^{-1/2} à D^{-1/2} H^l W^l ), where à = A + I adds self-loops, D is the degree matrix of Ã, H^l is the matrix of node features at layer l, W^l is a learnable weight matrix, and sigma is a nonlinearity. The input H^0 contains the initial node features.

Two main families exist: spectral GCNs based on graph signal processing with filters defined in the Fourier domain, and spatial GCNs that perform neighborhood aggregation directly in the graph. The Kipf and Welling 2017 simplification of the spectral approach popularized the first-order approximation and is widely used.

Applications include semi-supervised node classification on networks, graph classification, link prediction, and molecular property prediction. GCNs can scale to larger graphs with sampling and sparse computations, and have inspired many extensions such as Graph Attention Networks (GATs), GraphSAGE, and various normalization and residual strategies.

Limitations include over-smoothing with many layers, sensitivity to graph quality, and computational challenges on very large or dynamic graphs. Overall, GCNs provide a practical framework for learning from graph-structured data by combining feature information with graph topology.