How does mini batching help with dealing with variable sized input graphs

[VHK459]

I know that if input graph size is variable then we use minibatching to make a single giant block diagonal matrix consisting of all graphs.
But as show in this Graph Classification.ipynb example by pytorch-geometric the total size of each batch is not same,(as no. of graphs is same ) . So how is GCNConv handling this variability in input. As GCN is $\hat{X} = \tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}} X W$ where W is the projection matrix. As here size of input is variable then how does GCN use W matrix.

[rusty1s]

The weight matrix is independent of the size of nodes. It is just a matrix mapping a node feature to a transformed version of it, which is shared across all nodes. Does that answer your question?

[VHK459]

Ok, so we first calculate $\bar{X} = \tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}} X $ then multiply $\bar{X} W $ to get $ \hat{X}$ for all nodes and then do mean pooling to get graph embeddings. Correct ?

[rusty1s]

That's correct.