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\section{Consistency of notations}%
\label{sec:consistency}

Using the notation from the blog and the paper, including $H'\in\mathbb{R}^{N\times D}$, $A\in\mathbb{R}^{N\times N}$, $H\in\mathbb{R}^{N\times D}$, the propagation rule is:

\begin{equation}
	H' = A H
\end{equation}

is equivalent to

\begin{equation}
	\begin{split}
		H'_{v_i, m} & = \sum_l \overbrace{A_{v_i, l}}^{\in \{0, 1\}} H_{l, m} \\
		            & = \sum_{l'} H_{l',m}
	\end{split}
\end{equation}

where $l'$ takes all $A_{v_i,l}=1$, i.e.\ neighbours, into account.

Hence, I do understand that the notations are equal.

\section{Formulate normalisation}

Use from ICLR 2017 paper the normalisation $D_{ii} = \sum_j \hat{A}_{ij}$.

Use the knowledge from the blog post to write:

\begin{equation}
	\begin{split}
		\overbrace{H'}^{\in \mathbb{R}^{N\times D}} & = \overbrace{D^{-1}}^{\in\mathbb{R}^{N\times N}} \overbrace{\hat{A}}^{\in \mathbb{R}^{N\times N}} \overbrace{H}^{\in\mathbb{R}^{N\times D}} \\
		\Leftrightarrow H'_{v_i, m} & = \sum_k \overbrace{D^{-1}_{v_i,k}}^{=D_{v_i,v_i}, \delta_{v_i, k}} {(\hat{A} H)}_{km} \\
			& = D^{-1}_{v_i, v_i} {(\hat{A} H)}_{v_i, m} \\
			& = D^{-1}_{{v}_i, v_i} \sum_k \hat{A}_{v_i, k} H_{k, m} \\
			& = \sum_k \underbrace{\frac{\hat{A}_{v_i, k}}{\sum_j \hat{A}_{v_i, j}}}_{\text{normalised}} H_{k, m}.
	\end{split}
\end{equation}

The weighting by the adjacency matrix is, indeed, normalised to its column sums.

Note that this part is also normalised:

\begin{equation}
\begin{split}
	\sum_m H'_{v_i, m} & = \sum_m \sum_k \frac{\hat{A}_{v_i, k}}{\sum_j \hat{A}_{v_i, j}} H_{k, m} \\
	& = \sum_k \frac{\hat{A}_{v_i, k}}{\sum_j \hat{A}_{v_i, j}} \underbrace{\sum_m H_{k,m}}_{=1\text{, per construction}} \\
	& = 1
\end{split}
\end{equation}

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