Answers.Body: <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>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-05-03T05:56:16.063Z
Answers.LastActivityDate: 2022-05-03T12:42:55.743Z
Answers.LastEditDate: 2022-05-03T12:42:55.743Z
Answers.UserDisplayName: OmG
Body: <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>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-02T17:32:01.990Z
Comments.Text: Can you please use latex to transcribe the equation? You can use a tool like Mathpix Snipping tool to quickly convert text to mathjax.
Comments.UserDisplayName: nbro
Tags: <machine-learning><deep-learning><graph-neural-networks><gnn>
Title: How to understand the GCN equation?
UserDisplayName: Exploring
ThreadText: # 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.  
  
  ---
  
  ## 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: $\rho(\tilde{A}XW\_0)$.
  
  
  ---
  
  