If you read modern (that is, 2018-2020) papers using deep learning on molecular inputs, almost all of them use some variant of graph convolution. So, I decided to go back through the citation chain and read the earliest papers that thought to apply this technique to molecules, to get an idea of lineage of the technique within this domain. 

This 2015 paper, by Duvenaud et al, is the earliest one I can find. It focuses the entire paper on comparing differentiable, message-passing networks to the state of the art standard at the time, circular fingerprints (more on that in a bit). I really appreciated this approach, which, rather than trying to claim an unrealistic level of novelty, goes into detail on the prior approach, and carves out specific areas of difference. At a high level, the authors' claim is: our model is, in its simplest case, a more flexible and super version of existing work. The unspoken corollary, which ended up being proven true, is that the flexibility of the neural network structure makes it easy to go beyond this initial level of simplicity. 

Circular Fingerprinting (or, more properly, Extended-Connectivity Circular Fingerprints), is a fascinating algorithm that captures many of the elements of convolution: shared weights, a hierarchy of kernels that match patterns at different scales, and a clever way of aggregating information across an arbitrary number of input nodes. Mechanistically, Circular Fingerprints work by: 
1) Taking each atom, and creating a concatenated vector of its basic features, along with the basic features of each atom it's bonded to (with bonded neighbors quasi-randomly)

2) Calculating next-level features by applying some number of hash functions (roughly equivalent to convolutional kernels) to the neighborhood feature vector at the lower level to produce an integer 

3) For each feature, setting the value of the fingerprint vector to 1 at the index implied by the integer in step (2) 

4) Iterating this process at progressively higher layers, using the hash 

The effect of this is to assign each index of the vector to an binary feature (modulo hash collisions), where that feature is activated if an exact match is found to a structure within a given atom.  Its main downside is that (a) its "kernel" equivalents are fixed and not trainable, since they're just random hashes, and (b) its features represent *exact* matches to lower-level feature patterns, which means you can't have one feature activated to different degrees by variations on a pattern it's identifying.

https://i.imgur.com/V8FpfVE.png

Duvenaud et al present their alternative in terms of keeping a similar structure, but swapping out fixed and binary components for trainable (because differentiable) and continuous ones. Instead of concatenating a random sorting of atom neighbors to enforce invariance to sorting, they simply sum feature vectors across neighbors, which is also an order-invariantoperation. Instead of applying hash functions, they apply parametrized kernel functions, with the same parameters used across all aggregated neighborhood vectors . This will no longer look for exact matches, but will activate to the extent a structure within an atom matches against a kernel pattern. Then, these features are put into a softmax, which instead setting an index of a vector to a sharp one value, activates different feature indices in the final vector to differing degrees. The final fingerprint is simply the sum of these softmax feature activations for each atom. 

The authors do a few tests to confirm their substitution is working well, including starting out with a random network (to better approximate the random hash functions), comparing distances between atoms according to either the circular or neural footprint (which had a high correlation), and confirming that that performs similarly to circular fingerprints on a set of supervised learning tasks on modules. When they trained weights to be better than random on three such supervised tasks, they found that their model was comparable or better than circular fingerprints on all three (to break that down: it was basically equivalent on one, and notably better on the other two, according to mean squared error) 

This really is the simplest possible version of a message-passing or graph convolutional network (it doesn't use edge features, it doesn't calculate features of a neighbor-connection according to the features of each node, etc), but it's really satisfying to see it laid out as a next-step alternative that offered value just by stepping away from exact-match feature dynamics and non-random functions, even without all the sophisticated additions that would later be added to such models.