<p><span class="math-container">$\tilde{A}$</span> is related to <a href="https://en.wikipedia.org/wiki/Laplacian_matrix" rel="nofollow noreferrer">normalized Laplacian matrix</a> that &quot;shows many useful properties&quot; of matrix <span class="math-container">$A$</span>. Note that:</p>
<blockquote>
<p>Since the degree matrix <span class="math-container">$D$</span> is diagonal, its reciprocal square root <span class="math-container">$D^{-{\frac {1}{2}}}$</span> is just the diagonal matrix whose diagonal entries are the reciprocals of the square roots of the diagonal entries of <span class="math-container">$D$</span>. 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.</p>
</blockquote>
<p>Hence, if <span class="math-container">$A$</span> represents the adjacency matrix, first normalize it by degree matrix and get <span class="math-container">$\tilde{A}$</span>. Then, times the result into the feature matrix <span class="math-container">$X$</span> (in the first layer) and get the output times the weight of the network, denoted by <span class="math-container">$W_0$</span>. In the end, apply the softmax function on the the obtained result to get the final output of this layer: <span class="math-container">$\rho(\tilde{A}XW_0)$</span>.</p>

CC BY-SA 4.0
2022-05-03T05:56:16.063Z
2022-05-03T12:42:55.743Z
2022-05-03T12:42:55.743Z
OmG
<p>I understand GCN does message passing with its neighbours to learn the node embedding.</p>
<p>But I don't understand the following equation.</p>
<p>What &quot;tilda&quot; is referring to equation <code>1</code>?</p>
<p><a href="https://i.sstatic.net/GewBb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GewBb.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-05-02T17:32:01.990Z
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
<machine-learning><deep-learning><graph-neural-networks><gnn>
How to understand the GCN equation?
Exploring
# How to understand the GCN equation?

(Asked by: username_0 on 2022-05-02)

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](https://i.sstatic.net/GewBb.png)](https://i.sstatic.net/GewBb.png)

**Tags:** machine-learning, deep-learning, graph-neural-networks, gnn

> **Comments:**
> **username_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.  

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## 1 answers

### Answer by username_2 on 2022-05-03. Score: 1

$\tilde{A}$ is related to [normalized Laplacian matrix](https://en.wikipedia.org/wiki/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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