# Graph Neural Networks: Quesitons about different GCN Architectures

This might be moreof a question about nested function classes:

For k class node classification in a graph with n nodes, and d feature vector. I want to compare

Architecture I: the GCN model of Kipf/ Welling with two graph convolutional layers: $$\mathbf{Y}=\operatorname{softmax}\left(\mathbf{A} \xi\left(\mathbf{A X W}_{1}\right) \mathbf{W}_{2}\right)$$

where $$\mathbf{X}$$ is $$n \times d,$$ $$\mathbf{Y}$$ is $$n \times k, \mathbf{A}$$ is a fixed $$n \times n$$ graph diffusion matrix, and $$\mathbf{W}_{1}, \mathbf{W}_{2}$$ are learnable weight matrices of size $$d \times d^{\prime}$$ and $$d^{\prime} \times 2,$$ respectively, shared across all nodes, and $$\xi$$ is a nonlinearity.

Architecture II: a single-layer graph neural network of the form: $$\mathbf{Y}=\operatorname{softmax}\left(\mathbf{A}^{2} \mathbf{X W}\right)$$ where $$\mathbf{W}$$ is a learnable weight matrix of size $$d \times 2$$.

Now I'm wondering

• $$\operatorname{Can} \xi, d^{\prime}$$ be chosen in a way that both architectures have the same expressive power? (i.e. can represent the same class of functions)?

• $$\operatorname{Can} \xi, d^{\prime}$$ be chosen in a way that Architecture II is more expressive?

• What would be the advantage in training complexity of Architecture II when applied to large-scale graphs.

• Could you please focus on 1 specific question (eventually, you can add 1-2 more sub-questions, provided they are sub-questions and not distinct questions)? Then could you please put that specific question in the title? To me, it seems that your questions are quite distinct (and a bit broad) and require a lot of effort to answer properly. So, I actually suggest that you focus on 1 of those questions and remove the others. If you have more questions, ask them in other separate posts.
– nbro
Jan 12 at 20:31