This paper was published after the 2015 Duvenaud et al paper proposing a differentiable alternative to circular fingerprints of molecules: substituting out exact-match random hash functions to identify molecular structures with learned convolutional-esque kernels. As far as I can tell, the Duvenaud paper was the first to propose something we might today recognize as graph convolutions on atoms. I hoped this paper would build on that one, but it seems to be coming from a conceptually different direction, and it seems like it was more or less contemporaneous, for all that it was released later. 

This paper introduces a structure that allows for more explicit message passing along bonds, by calculating atom features as a function of their incoming bonds, and then bond features as a function of their constituent atoms, and iterating this procedure, so information from an atom can be passed into a bond, then, on the next iteration, pulled in by another atom on the other end of that bond, and then pulled into that atom's bonds, and so forth. This has the effect of, similar to a convolutional or recurrent network, creating representations for each atom in the molecular graph that are informed by context elsewhere in the graph, to different degrees depending on distance from that atom. 

More specifically, it defines: 

- A function mapping from a prior layer atom representation to a subsequent layer atom representation, taking into account only information from that atom (Atom to Atom)
- A function mapping from a prior layer bond representation (indexed by the two atoms on either side of the bond), taking into account only information from that bond at the prior layer (Bond to Bond)
- A function creating a bond representation by applying a shared function to the atoms at either end of it, and then combining those representations with an aggregator function (Atoms to Bond)
- A function creating an atom representation by applying a shared function all the bonds that atom is a part of, and then combining those results with an aggregator function (Bonds to Atom)

At the top of this set of layers, when each atom has had information diffused into it by other parts of the graph, depending on the network depth, the authors aggregate the per-atom representations into histograms (basically, instead of summing or max-pooling feature-wise, creating course distributions of each feature), and use that for supervised tasks. 

One frustration I had with this paper is that it doesn't do a great job of highlighting its differences with and advantages over prior work; in particular, I think it doesn't do a very good job arguing that its performance is superior to the earlier Duvenaud work. That said, for all that the presentation wasn't ideal, the idea of message-passing is an important one in graph convolutions, and will end up becoming more standard in later works.