From 0e9e33a28195f8d651ef223334c2627e53f39931 Mon Sep 17 00:00:00 2001
From: chris-admin <chricchio.risi@icloud.com>
Date: Thu, 11 Sep 2025 10:02:02 +0200
Subject: [PATCH] Added Introduction about GNNs

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 Chapters/14-GNN-GCN/INDEX.md | 91 ++++++++++++++++++++++++++++++++++++
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+# Graph ML
+
+## Graph Introduction
+
+- **Nodes**: Pieces of Information
+- **Edges**: Relationship between nodes
+    - **Mutual**
+    - **One-Sided**
+- **Directionality**
+    - **Directed**: We care about the order of connections
+        - **Unidirectional**
+        - **Bidirectional**
+    - **Undirected**: We don't care about order of connections
+
+Now, we can have attributes over
+
+- **nodes**
+- **edges**
+- **master nodes** (a collection of nodes and edges)
+
+for example images may be represented as a graph where each non edge pixel is a vertex connected to other 8 ones.
+Its information at the vertex is a 3 (or 4) dimensional vector (think of RGB and RGBA)
+
+### Adjacency Graph
+
+Take a picture and make a matrix with dimension $\{0, 1\}^{(h \cdot w) \times (h \cdot w)}$ and we put a 1 if these
+nodes are connected (share and edge), or 0 if they do not.
+
+> [!NOTE]
+> For a $300 \times 250$ image our matrix would be $\{0, 1\}^{(250 \cdot 300) \times (250 \cdot 300)}$
+
+The way we put a 1  or a 0 has this rules:
+    - **Row element** has connection **towards** **Column element**
+    - **Column element** has a connection **coming** from **Row element**
+
+### Tasks
+
+#### Graph-Level
+
+We want to predict a graph property
+
+#### Node-Level
+
+We want to predict a node property, such as classification
+
+#### Edge-Level
+
+We want to predict relationships between nodes such as if they share an edge, or the value of the edge they share.
+
+For this task we may start with a fully connected graph and then prune edges, as predictions go on, to come to a
+sparse graph
+
+### Downsides of Graphs
+
+- They are not consistent in their structure and sometimes representing something as a graph is difficult
+- If we don't care about order of nodes, we need to find a way to represent this **node-order equivariance**
+- Graphs may be too large
+
+## Representing Graphs
+
+### Adjacency List
+
+We store info about:
+
+- **Nodes**: list of values. index $Node_k$ is the value of that node
+- **Edges**: list of values. index $Edge_k$ is the value of that edge
+- **Adjacent_list**: list of Tuples with indices over nodes. index $Tuple_k$
+    represent the Nodes involved in the $kth$ edge
+- **Graph**: Value of graph
+
+```python
+nodes: list[any] = [
+    "forchetta", "spaghetti", "coltello", "cucchiao", "brodo"
+]
+
+edges: list[any] = [
+    "serve per mangiare", "strumento", "cibo",
+    "strumento", "strumento", "serve per mangiare"
+]
+
+adj_list: list[(int, int)] = [
+    (0, 1), (0, 2), (1, 4),
+    (0, 3), (2, 3), (3, 4)
+]
+
+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)