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## Graph Neural Networks
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* Graph Neural Networks (GNNs): Neural Networks that operate on graph structured data
* Graph: consists of nodes that can be connected by edges, edges can be directed or undirected
* no grid structure as given for CNNs
* node features and edge features possible
* relation often represented by adjacency matrix: $A_{ij}=1$ if there is a link between node $i$ and $j$, else 0
* tasks on node level, edge level and graph level
* full lecture: \url{https://web.stanford.edu/class/cs224w/}
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\begin{center}
\includegraphics[width=1.1\textwidth]{figures/graph_example.png}
\normalsize
\end{center}
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## Simple Example: Zachary's karate club
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* link: \url{https://en.wikipedia.org/wiki/Zachary's_karate_club}
* 34 nodes: each node represents a member of the karate club
* 4 classes: a community each member belongs to
* task: classify the nodes
* many real world problems for GNNs exist, e.g.\ social networks, molecules, recommender systems, particle tracks
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\begin{center}
\includegraphics[width=1.\textwidth]{figures/karateclub.png}
\normalsize
\end{center}
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## From CNN to GNN
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/fromCNNtoGNN.png}
\normalsize
\newline
\tiny (from Stanford GNN lecture)
\end{center}
\normalsize
* GNN: Generalization of convolutional neural network
* No grid structure, arbitrary number of neighbors defined by adjacency matrix
* Operations pass information from neighborhood
## Architecture: Graph Convolutional Network
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* Message passing from connected nodes
* The graph convolution is defined as:
$$ H^{(l+1)} = \sigma \left( \tilde{D}^{\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}} H^{(l)} W^{(l)} \right)$$
* The adjacency matrix $A$ including self-connections is given by $\tilde{A}$
* The degree matrix of the corrected adjacency matrix is given by $\tilde{D}_{ii} = \Sigma_j \tilde{A}_{ij}$
* The weights of the given layer are called $W^{(l)}$
* $H^{(l)}$ is the matrix for activations in layer $l$
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\begin{center}
\includegraphics[width=1.1\textwidth]{figures/GCN.png}
\normalsize
\end{center}
\tiny \url{https://arxiv.org/abs/1609.02907}
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## Architecture: Graph Attention Network
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* Calculate the attention coefficients $e_{ij}$ from the features $\vec{h}$ for each node $i$ with its neighbors $j$
$$ e_{ij} = a\left( W\vec{h}_i, W\vec{h}_j \right)$$
$a$: learnable weight vector
* Normalize attention coefficients
$$ \alpha_{ij} = \text{softmax}_j(e_{ij}) = \frac{\text{exp}(e_{ij})}{\Sigma_k \text{exp}(e_{ik})} $$
* Calculate node features
$$
\vec{h}^{(l+1)}_i = \sigma \left( \Sigma \alpha_{ij} W \vec{h}^l_j \right)$$
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\begin{center}
\includegraphics[width=1.1\textwidth]{figures/GraphAttention.png}
\normalsize
\end{center}
\tiny \url{https://arxiv.org/abs/1710.10903}
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## Example: Identification of inelastic interactions in TRD
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* Identification of inelastic interactions of light antinuclei
in the Transition Radiation Detector in ALICE
* Thesis: \url{https://www.physi.uni-heidelberg.de/Publications/Bachelor_Thesis_Maximilian_Hammermann.pdf}
* Construct nearest neighbor graph from signals in detector
* Use global pooling for graph classification
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interaction of antideuteron:
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/antideuteronsgnMax.png}
\normalsize
\end{center}
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\begin{center}
\includegraphics[width=0.9\textwidth]{figures/GNN_conf.png}
\normalsize
\end{center}
## Example: Google Maps
* link: \url{https://www.deepmind.com/blog/traffic-prediction-with-advanced-graph-neural-networks}
* GNNs are used for traffic predictions and estimated times of arrival (ETAs)
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/GNNgooglemaps.png}
\normalsize
\end{center}
## Example: Alpha Fold
* link: \url{https://www.deepmind.com/blog/alphafold-a-solution-to-a-50-year-old-grand-challenge-in-biology}
* "A folded protein can be thought of as a 'spatial graph', where residues are the nodes and edges connect the residues in close proximity"
\begin{center}
\includegraphics[width=0.9\textwidth]{figures/alphafold.png}
\normalsize
\end{center}
## Exercise 1: Illustration of Graphs and Graph Neural Networks
On the PyTorch webpage, you can find official examples for the application of Graph Neural Networks:
https://pytorch-geometric.readthedocs.io/en/latest/get_started/colabs.html
\vspace{3ex}
The first introduction notebook shows the functionality of graphs with the example of the Karate Club. Follow and reproduce the first [\textcolor{green}{notebook}](https://colab.research.google.com/drive/1h3-vJGRVloF5zStxL5I0rSy4ZUPNsjy8?usp=sharing). Study and understand the data format.
\vspace{3ex}
At the end, the separation power of Graph Convolutional Networks (GCN) are shown via the node embeddings. You can replace the GCN with a Graph Attention Layers and compare the results.