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I have only just started to look into graph neural networks and I am a little confused on the node embedding process. Here is my understanding, please let me know if i misunderstood:

Given unlabelled data, we try to construct a graph $G=(V,E)$ and generate a mask that contains important features of each node such as shape, size, intensity etc. Then let's say the aim is to try and classify the nodes. So, this is where we use node embedding, it seems that it takes all of those aforementioned features and tries to condense them into a embedding matrix. And the reason we do this is because this removes all unneccessary information from the adjacency matrix and only contains all the important features of the nodes. Then afterwards we use this matrix with graph convolutions to ultimately classify unseen images. Is this right?

So let's take an example, I've been looking into GraphSage, as this seems one of the best methods (is it?). It seems it can learn embeddings from a local neighbourhood, but i want to know how does that work?

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First, let me give you an intuitive explanation. When you drop a pebble in water, you see the ripples being formed. Imagine that in reverse. All those ripples comes together at the point from which they started.

Node embeddings are like that. You take the information of neighbourhood nodes and combine it with the information in the original node. The art lies in how you combine the neighbourhood nodes and until which k-hop neighbours you do it. 1-hop means direct neighbours, 2-hop means neighbour of neighbours and so on.

In context of GraphSAGE, the combination of information is done using a simple feedforward neural network and the neighbourhood is incorporated iteratively. See link for further understanding.

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