Added Polynomials of Laplacian

Chapters/14-GNN-GCN/INDEX.md

@@ -89,3 +89,110 @@ graph: any = "tavola"
 If we find some parts of the graph that are disconnected, we can just avoid storing and computing those parts
 
 ## Graph Neural Networks (GNNs)
+
+At the simpkest form we take a **graph-in** and **graph-out** approach with MLPs separate for
+vertices, edges and master nodes that we apply **one at a time** over each element
+
+$$
+\begin{aligned}
+    V_{i + 1} &= MLP_{V_{i}}(V_{i}) \\
+    E_{i + 1} &= MLP_{E_{i}}(E_{i}) \\
+    U_{i + 1} &= MLP_{U_{i}}(U_{i}) \\
+\end{aligned}
+$$
+
+### Pooling
+
+> [!CAUTION]
+> This step comes after the embedding phase described above
+
+This is a step that can be used to take info about other elements, different from what we were considering
+(for example, taking info from edges while making the computation over vertices).
+
+By using this approach we usually gather some info from edges of a vertex, then we concat them in a matrix and
+aggregate by summing them.
+
+### Message Passing
+
+Take all node embeddings that are in the neighbouroud and do similar steps as the pooling function.
+
+### Special Layers
+
+<!-- TODO: Read PDF 14 Anelli pg 47 to 52 -->
+
+## Polynomial Filters
+
+### Graph Laplacian
+
+Let's set an order over nodes of a graph, where $A$ is the adjacency matrix:
+
+$$
+
+D_{v,v} = \sum_{u} A_{v,u}
+
+$$
+
+In other words, $D_{v, v}$ is the number of nodes connected ot that one
+
+The **graph Laplacian** of the graph will be
+
+$$
+L = D - A
+$$
+
+### Polynomials of Laplacian
+
+These polynomials, which have the same dimensions of $L$, can be though as being **filter** like in
+[CNNs](./../7-Convolutional-Networks/INDEX.md#convolutional-networks)
+
+$$
+p_{\vec{w}}(L) = w_{0}I_{n} + w_{1}L^{1} + \dots + w_{d}L^{d} = \sum_{i=0}^{d} w_{i}L^{i}
+$$
+
+We then can get a ***filtered node*** by simply multiplying the polynomial with the node value
+
+$$
+\begin{aligned}
+    \vec{x}' = p_{\vec{w}}(L) \vec{x}
+\end{aligned}
+$$
+
+> [!NOTE]
+> In order to extract new features for a single vertex, supposing only $w_1 \neq 0$
+>
+> Observe that we are only taking $L_{v}$
+>
+> $$
+> \begin{aligned}
+>     \vec{x}'_{v} &= (L\vec{x})_{v}    \\
+>       &= \sum_{u \in G} L_{v,u} \vec{x}_{u}   \\
+>       &= \sum_{u \in G} (D_{v,u} - A_{v,u}) \vec{x}_{u}   \\
+>       &= \sum_{u \in G} D_{v,u} \vec{x}_{u} - A_{v,u} \vec{x}_{u} \\
+>       &= D_{v, v} \vec{x}_{v} - \sum_{u \in \mathcal{N}(v)} \vec{x}_{u}
+> \end{aligned}
+> $$
+>
+> Where the last step holds as $D$ is a diagonal matrix, and in the summatory we are only considering the neighbours
+> of v
+>
+> It can be demonstrated that in any graph
+>
+> $$
+> dist_{G}(v, u) > i \rightarrow L_{v, u}^{i} = 0
+> $$
+>
+> More in general it holds
+>
+> $$
+> \begin{aligned}
+>     \vec{x}'_{v} = (p_{\vec{w}}(L)\vec{x})_{v} &= (p_{\vec{w}}(L))_{v} \vec{x}   \\
+>       &= \sum_{i = 0}^{d} w_{i}L_{v}^{i} \vec{x}   \\
+>       &= \sum_{i = 0}^{d} w_{i} \sum_{u \in G} L_{v,u}^{i}\vec{x}_{u}   \\
+>       &= \sum_{i = 0}^{d} w_{i} \sum_{\substack{u \in G \\ dist_{G}(v, u) \leq i}} L_{v,u}^{i}\vec{x}_{u}   \\
+> \end{aligned}
+> $$
+>
+> So this shows that the degree of the polynomial decides the max number of hops
+> to be included during the filtering stage, like if it were defining a [kernel](./../7-Convolutional-Networks/INDEX.md#filters)
+
+### ChebNet