  In many domains data is currently represented as graphs and therefore, the
graph representation of this data becomes increasingly important in machine
learning. Network data is, implicitly or explicitly, always represented using a
graph shift operator (GSO) with the most common choices being the adjacency,
Laplacian matrices and their normalisations. In this paper, a novel
parametrised GSO (PGSO) is proposed, where specific parameter values result in
the most commonly used GSOs and message-passing operators in graph neural
network (GNN) frameworks. The PGSO is suggested as a replacement of the
standard GSOs that are used in state-of-the-art GNN architectures and the
optimisation of the PGSO parameters is seamlessly included in the model
training. It is proved that the PGSO has real eigenvalues and a set of real
eigenvectors independent of the parameter values and spectral bounds on the
PGSO are derived. PGSO parameters are shown to adapt to the sparsity of the
graph structure in a study on stochastic blockmodel networks, where they are
found to automatically replicate the GSO regularisation found in the
literature. On several real-world datasets the accuracy of state-of-the-art GNN
architectures is improved by the inclusion of the PGSO in both node- and
graph-classification tasks.
