In a graph convolutional network, we assume that the graph $G$ is generated wrt some observation noise. During learning, we make small random perturbations $ΔG$ of the graph and try to improve generalization.

Based on quantum information geometry, $ΔG$ can be characterized by the eigendecomposition of the graph Laplacian matrix. We try to minimize the loss wrt the perturbed $G+Δ{G}$ while making $Δ{G}$ to be effective in terms of the Fisher information of the neural network.

Our proposed model can consistently improve graph convolutional networks on semi-supervised node classification tasks with reasonable computational overhead. We present three different geometries on the manifold of graphs: the intrinsic geometry measures the information theoretic dynamics of a graph; the extrinsic geometry characterizes how such dynamics can affect externally a graph neural network; the embedding geometry is for measuring node embeddings.

These new analytical tools are useful in developing a good understanding of graph neural networks and fostering new techniques.