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In many real world applications, data are characterized by a complexstructure, that can be naturally encoded as a graph. In the last years, thepopularity of deep learning techniques has renewed the interest in neuralmodels able to process complex patterns. In particular, inspired by the GraphNeural Network (GNN) model, different architectures have been proposed toextend the original GNN scheme. GNNs exploit a set of state variables, eachassigned to a graph node, and a diffusion mechanism of the states amongneighbor nodes, to implement an iterative procedure to compute the fixed pointof the (learnable) state transition function. In this paper, we propose a novelapproach to the state computation and the learning algorithm for GNNs, based ona constraint optimisation task solved in the Lagrangian framework. The stateconvergence procedure is implicitly expressed by the constraint satisfactionmechanism and does not require a separate iterative phase for each epoch of thelearning procedure. In fact, the computational structure is based on the searchfor saddle points of the Lagrangian in the adjoint space composed of weights,neural outputs (node states), and Lagrange multipliers. The proposed approachis compared experimentally with other popular models for processing graphs.

A Lagrangian Approach to Information Propagation in Graph Neural Networks