How does graph classification work with graph neural networks
The equation you pointed out in the paper has posed the graph learning problem based only on the nodes of the graph. Therefore, to perform graph level tasks like graph classification, one would need a 'special node' which introduces a node that represents the entire graph. This is all just to make the equation hold for graph level tasks which are not dependent on just one node.

The paper which you are reading is quite old and the new GNN formulations are based on the kipf-GCN (https://arxiv.org/abs/1609.02907). It is much simpler to understand, implement and has become the standard GCN. If you want to keep with the current hot topic of GNN, I would recommend you having a look at Kipf's blog (https://tkipf.github.io/graph-convolutional-networks/) for a simple intuition for the modern-day GCNs.