Message-passing has proved to be an effective way to design graph neural\nnetworks, as it is able to leverage both permutation equivariance and an\ninductive bias towards learning local structures in order to achieve good\ngeneralization. However, current message-passing architectures have a limited\nrepresentation power and fail to learn basic topological properties of graphs.\nWe address this problem and propose a powerful and equivariant message-passing\nframework based on two ideas: first, we propagate a one-hot encoding of the\nnodes, in addition to the features, in order to learn a local context matrix\naround each node. This matrix contains rich local information about both\nfeatures and topology and can eventually be pooled to build node\nrepresentations. Second, we propose methods for the parametrization of the\nmessage and update functions that ensure permutation equivariance. Having a\nrepresentation that is independent of the specific choice of the one-hot\nencoding permits inductive reasoning and leads to better generalization\nproperties. Experimentally, our model can predict various graph topological\nproperties on synthetic data more accurately than previous methods and achieves\nstate-of-the-art results on molecular graph regression on the ZINC dataset.\n