# How to understand the GCN equation?

I understand GCN does message passing with its neighbours to learn the node embedding.

But I don't understand the following equation.

What "tilda" is referring to equation 1? • Can you please use latex to transcribe the equation? You can use a tool like Mathpix Snipping tool to quickly convert text to mathjax.
– nbro
May 2, 2022 at 17:32

$$\tilde{A}$$ is related to normalized Laplacian matrix that "shows many useful properties" of matrix $$A$$. Note that:
Since the degree matrix $$D$$ is diagonal, its reciprocal square root $$D^{-{\frac {1}{2}}}$$ is just the diagonal matrix whose diagonal entries are the reciprocals of the square roots of the diagonal entries of $$D$$. If all the edge weights are nonnegative then all the degree values are automatically also nonnegative and so every degree value has a unique positive square root. To avoid the division by zero, vertices with zero degrees are excluded from the process of the normalization.
Hence, if $$A$$ represents the adjacency matrix, first normalize it by degree matrix and get $$\tilde{A}$$. Then, times the result into the feature matrix $$X$$ (in the first layer) and get the output times the weight of the network, denoted by $$W_0$$. In the end, apply the softmax function on the the obtained result to get the final output of this layer: $$\rho(\tilde{A}XW_0)$$.