Graph Neural Networks (GNNs) are a powerful representational tool for solving problems on graph-structured inputs. In almost all cases so far, however, they have been applied to directly recovering a final solution from raw inputs, without explicit guidance on how to structure their problem-solving. Here, instead, we focus on learning in the space of algorithms: we train several state-of-the-art GNN architectures to imitate individual steps of classical graph algorithms, parallel (breadth-first search, Bellman-Ford) as well as sequential (Prim's algorithm). As graph algorithms usually rely on making discrete decisions within neighbourhoods, we hypothesise that maximisation-based message passing neural networks are bestsuited for such objectives, and validate this claim empirically. We also demonstrate how learning in the space of algorithms can yield new opportunities for positive transfer between tasks-showing how learning a shortest-path algorithm can be substantially improved when simultaneously learning a reachability algorithm.
This paper investigates using GNNs to learn graph algorithms. It proposes a model which consists of algorithm-dependent encoder and decoder, and algorithm-independent processor. 
Authors try to learn BFS, Bellman-Ford and Prim algorithms on various types of random graphs. 
Experimental results suggest that MPNN with max aggregator outperforms other variants significantly in terms of generalization. 
Overall, this paper presents a solid contribution on learning graph algorithms using GNNs, despite the caveat for some clarifications on the model and experiments. 
Given these clarifications in an author response, I would be willing to increase the score.
Pros:
1, Although it is less satisfying to learn to solve graph problems where polynomial-time algorithms exist, I still appreciate the contribution of the paper, especially the algorithm-independent processor. It is one step forward to models which could learn meta-level representation of algorithms. The findings of this paper may suggest that different types of operators used in the GNN may have different inductive bias in learning different types of algorithms.
2, Experimental comparisons are adequate and convincing. The detailed analysis of empirical results also provide good explanation.
Cons & Suggestions:
1, Given the dynamic programming nature of the most of tasks, it is not that surprising that MPNN with max aggregation could solve them pretty well. What surprises me is that MPNN-mean / MPNN-sum could not generalize well (i.e., performance drops significantly on 50 and 100 nodes settings) on the reachability task. In my opinion, the reachability task could be easily handled by any diffusion/propagation based models including, e.g., MPNN, GCN, GAT, as long as the information is spread over the input graph. Could you explain why does this happen? 
2, Training details are sparse. If I understood correctly, the training of various GNNs is done by teacher forcing such that each step some supervised information collected from the underlying graph algorithms is provided to GNNs. However, it is not clear that what supervised information is exactly provided under each graph algorithm. It would be great to have a table to summarize what the input and supervised output information is for each graph algorithm. 
3, The processor is trained to learn multiple graph algorithms simultaneously. A natural question to ask is how does the performance change when you train the processor with different combinations of algorithms? What is the impact of correlations between different graph algorithms? Did you explore learning with graph algorithms following some sequential schedule, e.g., curriculum learning?
4, It would be great to provide more results on larger size graphs, e.g., trained on 20 nodes but test on 1000 or more nodes. I saw one set of experiments trained on 100 and test 1000 nodes in the appendix. More results along this line would make the paper more convincing on the generalization.
5, It would be great to discuss [1] as it also studies how to use GNNs to learn a special class of graph algorithms, i.e., probabilistic inference algorithms on graphs. It is shown in [1] that belief propagation algorithm could be seen as a specially constructed GNN. Interestingly, some well-known graph algorithms could be re-formulated as probabilistic inference algorithms, e.g., graph-cut could be reformulated as max-product [2].
[1] Yoon, K., Liao, R., Xiong, Y., Zhang, L., Fetaya, E., Urtasun, R., Zemel, R. and Pitkow, X., 2018. Inference in probabilistic graphical models by graph neural networks. arXiv preprint arXiv:1803.07710.
[2] Tarlow, D., Givoni, I.E., Zemel, R.S. and Frey, B.J., 2011, July. Graph Cuts is a Max-Product Algorithm. In UAI (pp. 671-680).
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Thanks for the thorough response! It resolves my concerns. I improved my score.