Before revision:  Since none of the classical numerical graph invariants by itself seems to allow to single out the best networks , we introduced new numerical characteristics that selected a set of quasi-1-dimensional graphs, which were the majority among the best performing networks. 
After revision:  Since none of the classical numerical graph invariants by itself seems to allow to single out the best networks , we introduced new numerical characteristics that selected a set of quasi-1-dimensional graphs, which were the majority among the best performing networks. We also find that networks with primarily short-range connections perform better than networks which allow for many long-range connections. Moreover, many resolution reducing pathways are beneficial. We provide a dataset of 1020 graphs and the test accuracies of their corresponding neural networks at URL