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In the basic variant of GCN we have the following:

enter image description here

Here we aggregate the information from the adjacent node and pass it to a neural network, then transform our own information and add them all.

But the main question is: how can we ensure that $W_{k}(\sum(\frac{h_k}{N(V)})$ will be the same size as $B_{k}h_{v}$ and does $B_{k}$ emply another neural network?

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I think the picture you're presenting is mostly for educational purposes and that's why they are excluding the node itself from it's neighbors and using two distinct networks (most of the papers I've read they are using the same network for the neighbors and for the center node). But you are right the two networks needs to have the same input and output shapes otherwise the point-wise summation between the two terms is not possible.

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  • $\begingroup$ you said that "I think the picture you're presenting is mostly for educational purposes and that's why they are excluding the node itself from it's neighbors". I do not understand the connection between educational purpose and excluding the node itself from it's neighbors. $\endgroup$
    – Swakshar Deb
    Jul 23, 2020 at 14:19
  • $\begingroup$ Most of the paper I've read are defining the computation of GCN as follows H_l = AXW. Where A is the adjacency matrix X is the input (or previous layer embedding) and W are the parameters to be learned. You can see how such an definition is not as user friendly as the one you presented $\endgroup$
    – razvanc92
    Jul 23, 2020 at 20:15

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