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Recently, neural networks have been generalized to process data on graphs, with cutting-edge results in traditional tasks such as node classification and link prediction. These methods have all been formulated in a way suited only to data on the nodes of a graph, based on spectral graph theory. Using tools from algebraic topology, it is possible to reason about oriented data on higher-order structures by relying on the so-called Hodge Laplacian. Our goal is to develop techniques for applying the Hodge Laplacian to process data on higher-order graph structures using graph neural networks. To illustrate the practical value of this framework, we tackle the problem of flow interpolation: Given observations of flow over a subset of the edges of a graph, how can flow over the unobserved edges be inferred? We propose an architecture based on recurrent neural networks for performing flow interpolation, and demonstrate it on urban traffic data.
