How does Weight Sharing with the Generalization in Graph Neural Networks work?

Question

I have two closely related points regarding the weight sharing and generalization of graph Neural network. For illustration purposes, I attached two images which I reference. Images are taken from the Stanford course "CS224W:Machine Learning with Graphs" given by "Jure Leskovec"

1. In the centre above one can see that the blue, red and green node feed into the yellow one. The same can be seen one the right side with the same colour configuration. Are now these parameters shared across all computational graphs which 3 nodes feeding into 1?

2. If yes, then how does this generalize to a potential new graph? In the second image above, one can see that there are new types of computational graphs needed, e.g. on the right side the orange node has five neighbours while there is no such node on the left side. So, how does the generalization work in this case?

Answer 1

By my understanding, describing now the images: Compute graph for node A and B

At the bottom each node depicted has some initial input feature vector. This vector gets then multiplied by some matrix creating a new, more abstract feature vector. This corresponds to the first graph convolutional layer. The number of these new features is a hyperparameter and solely dependent on the matrix. Since every node in this specific has the same feature vectors this is well defined. The number of nodes connected with the next upper node is solely impacting the aggregation part, the more connected nodes simply the more things to add. So, it seems like I had misunderstood this all along and hopefully this helps some other people in the future.
This can of course we be written more clearly/concise, but after 7 months without answer, there doesn't seem a hurry. I leave that to do for later.