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Graph algorithms are key tools in many fields of science and technology. Someof these algorithms depend on propagating information between distant nodes ina graph. Recently, there have been a number of deep learning architecturesproposed to learn on undirected graphs. However, most of these architecturesaggregate information in the local neighborhood of a node, and therefore theymay not be capable of efficiently propagating long-range information. To solvethis problem we examine a recently proposed architecture, wave, whichpropagates information back and forth across an undirected graph in waves ofnonlinear computation. We compare wave to graph convolution, an architecturebased on local aggregation, and find that wave learns three differentgraph-based tasks with greater efficiency and accuracy. These three tasksinclude (1) labeling a path connecting two nodes in a graph, (2) solving a mazepresented as an image, and (3) computing voltages in a circuit. These tasksrange from trivial to very difficult, but wave can extrapolate from smalltraining examples to much larger testing examples. These results show that wavemay be able to efficiently solve a wide range of problems that requirelong-range information propagation across undirected graphs. An implementationof the wave network, and example code for the maze problem are included in thetflon deep learning toolkit (https://bitbucket.org/mkmatlock/tflon).

Deep learning long-range information in undirected graphs with wave networks