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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?

enter image description here

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    $\begingroup$ Can you please use latex to transcribe the equation? You can use a tool like Mathpix Snipping tool to quickly convert text to mathjax. $\endgroup$
    – nbro
    May 2, 2022 at 17:32

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$\tilde{A}$ 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: $\rho(\tilde{A}XW_0)$.

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