explain the math of graph neural networks MLP in details and first principles

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Welcome to our exploration of Graph Neural Networks. We'll start by understanding how they extend traditional Multi-Layer Perceptrons to handle graph-structured data. A standard MLP processes fixed-size vectors through layers of neurons, where each layer applies a weight matrix, adds a bias, and applies a non-linear activation function. This architecture works well for many tasks but has limitations when dealing with graph data where relationships between elements matter. MLPs face three key limitations when dealing with graph data. First, they require fixed-size inputs, but graphs can have varying numbers of nodes and edges. Second, MLPs are not permutation invariant - if we reorder the nodes in our representation, the MLP will produce different outputs even though the graph structure remains the same. Third, MLPs cannot naturally leverage the graph structure to capture relationships between nodes. These limitations make standard MLPs unsuitable for directly processing graph-structured data without significant information loss. Graph Neural Networks solve the limitations of MLPs through a process called message passing. This framework has two key steps: First, the aggregation step, where each node collects information from its neighbors. This must be permutation invariant, meaning the order of neighbors doesn't matter. Common aggregation functions include sum, mean, or max. Second, the update step combines the aggregated neighbor information with the node's own current representation. This often uses MLP-like operations with weight matrices and non-linear activations. By stacking multiple GNN layers, nodes can incorporate information from neighbors further away in the graph. A complete GNN architecture consists of multiple message passing layers stacked together. Each layer allows nodes to gather information from progressively larger neighborhoods. The first layer captures immediate neighbors (1-hop), the second layer incorporates neighbors of neighbors (2-hop), and so on. After K layers, each node's representation contains information from its K-hop neighborhood. These learned node representations can then be used for various downstream tasks: For node classification, we apply a simple classifier to each node's final representation. For link prediction, we combine pairs of node representations to predict if an edge exists between them. For graph classification, we use a readout function to aggregate all node representations into a graph-level representation, which is then passed to a classifier. To summarize what we've learned about Graph Neural Networks: GNNs extend the principles of MLPs to handle graph-structured data by incorporating neighborhood information through message passing. This process combines permutation invariant aggregation with MLP-like update operations. By stacking multiple GNN layers, nodes can capture information from progressively larger neighborhoods. Unlike standard MLPs, GNNs preserve the graph structure and relationships between nodes, making them suitable for tasks where connectivity matters. Applications include node classification, link prediction, and graph-level tasks. The key difference is that while MLPs require fixed-size inputs and process samples independently, GNNs can handle variable-sized graphs and explicitly model relationships between elements.