Many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. In this paper, we propose a new neural network model, called graph neural network (GNN) model, that extends existing neural network methods for processing the data represented in graph domains. This GNN model, which can directly process most of the practically useful types of graphs, e.g., acyclic, cyclic, directed, and undirected, implements a function tau(G,n) isin IR m that maps a graph G and one of its nodes n into an m-dimensional Euclidean space. A supervised learning algorithm is derived to estimate the parameters of the proposed GNN model. The computational cost of the proposed algorithm is also considered. Some experimental results are shown to validate the proposed learning algorithm, and to demonstrate its generalization capabilities. Disciplines Physical Sciences and Mathematics

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In this paper, we will consider the approximation properties of a recently introduced neural network model called graph neural network (GNN), which can be used to process-structured data inputs, e.g., acyclic graphs, cyclic graphs, and directed or undirected graphs. This class of neural networks implements a function tau(G,n) is an element of IR(m) that maps a graph G and one of its nodes n onto an m-dimensional Euclidean space. We characterize the functions that can be approximated by GNNs, in probability, up to any prescribed degree of precision. This set contains the maps that satisfy a property called preservation of the unfolding equivalence, and includes most of the practically useful functions on graphs; the only known exception is when the input graph contains particular patterns of symmetries when unfolding equivalence may not be preserved. The result can be considered an extension of the universal approximation property established for the classic feedforward neural networks (FNNs). Some experimental examples are used to show the computational capabilities of the proposed model.

In the last decade, connectionist models have been proposed that can process structured information directly. These methods, which are based on the use of graphs for the representation of the data and the relationships within the data, are particularly suitable for handling relational learning tasks. In this paper, two recently proposed architectures of this kind, i.e. Graph Neural Networks (GNNs) and Relational Neural Networks (RelNNs), are compared and discussed, along with their corresponding learning schemes. The goal is to evaluate the performance of these methods on benchmarks that are commonly used by the relational learning community. Moreover, we also aim at reporting differences in the behavior of the two models, in order to gain insights on possible extensions of the approaches. Since RelNNs have been developed with the specific task of learning aggregate functions in mind, some experiments are run considering that particular task. In addition, we carry out more general experiments on the mutagenesis and the biodegradability datasets, on which several other relational learners have been evaluated. The experimental results are promising and suggest that RelNNs and GNNs can be a viable approach for learning on relational data.