Analysis of Graph Neural Networks with Theory of Markov Chains

In this paper, we provide a theoretical tool for the interpretation and analysis of \emph{graph neural networks} (GNNs). We use Markov chains on graphs to mathematically model the forward propagation processes of GNNs. The graph neural networks are divided into two classes of operator-consistent and operator-inconsistent based on whether the Markov chains are time-homogeneous. Based on this, we study \emph{over-smoothing} which is an important problem in GNN research. We attribute the over-smoothing problem to the convergence of an arbitrary initial distribution to a stationary distribution. We prove the effectiveness of the previous methods for alleviating the over-smoothing problem. Further, we give the conclusion that operator-consistent GNN cannot avoid over-smoothing at an exponential rate in the Markovian sense. For operator-inconsistent GNN, we theoretically give a sufficient condition for avoiding over-smoothing. Based on this condition, we propose a regularization term which can be flexibly added to the training of the neural network. Finally, we design experiments to verify the effectiveness of this condition. Results show that our proposed sufficient condition not only improves the performance but also alleviates the over-smoothing phenomenon.