  In a graph convolutional network, we assume that the graph $G$ is generated
wrt some observation noise. During learning, we make small random perturbations
$\Delta{}G$ of the graph and try to improve generalization. Based on quantum
information geometry, $\Delta{}G$ can be characterized by the
eigendecomposition of the graph Laplacian matrix. We try to minimize the loss
wrt the perturbed $G+\Delta{G}$ while making $\Delta{G}$ 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.
