Message propagation and aggregation

A GNN layer performs an aggregation of the messages sent by the neighbors of each node. Let's take a look at the following diagram:

Consider the question:

How is the embedding of node A in the third layer, L₃, computed?

By looking at the graph structure on the right-hand side of the diagram, you can see that A's neighbors are nodes B and D, hence the embedding h_A³ of node A after layer three is a combination of the embedding of nodes B and D in the previous layer, L₂. The embedding of B and D in layer two is similarly computed using the embedding of their neighbors in the previous layer, L₁. This is summarized by the following equation, giving the embedding of node i in the (k+1)^th layer:

h_i^k+1 = σ( W_k Σ_j h_j^k + B_k h_i^k)

The second term of the equation is there to make sure the representation of i in the k^th layer is also taken into account and σ is an activation function.

The most important thing to remember from this...