Yann Dubois explained very well about the "Interpretation of Symmetric Normalised Graph Adjacency Matrix?".
He defined that adjacency matrix  can be weighted. He also defined that D̂ is the degree matrix (diagonal matrix with number of neighbours on the diagonal) of Â. But in GCN, D̂ is calculated by summing the values of each row.
Since in GCN paper the adjacency matrix is not weighted, it means that the diagonal values are the number of neighbours for each node. But if we use the same formula in GCN for a weighted adjacency matrix the values are not the number of neighbours, and they are the summation of weights for each node.
In my case, normalizing by the number of neighbours works properly. But when I use summation of weights to calculate D̂, my model does not work properly. My question is that: Is it correct if I use the number of neighbours for a weighted graph, instead of summing the weights? Could you please explain in detail the intuition and math behind it?
Thanks in advance.
