## 2 Materials and Methods
### Methodology
#### 2.2.3 The Graph Network Architecture  
Figure 5 gives the pipeline of the OGE architecture. Its basic components include a GAT-dominated encoder, a GCN-dominated decoder, the loss function, and so on. A detailed description of them is provided below. (1) The GAT-Dominated Encoder. GAT is one of the most popular GNN (Graph Neural Network) architectures and is considered as the state-of-the-art architecture for representation learning with graphs [33]. As illustrated in Figure 6, GAT is a two-step procedure: first, it calculates the attention coefficients (or say weights) of each graph edge according to the masked attention mechanism; afterwards, it aggregates the neighboring node features according to their weights [27]. Figure 4: The geochemical topology graph in the study area. Red dot—the graph node; black line—edge. Figure 5: The overall OGE framework for geochemical anomaly identification. GATv2: Graph Attention Networks (Version 2); GCN: Graph Convolution Networks; D_in/D_out: Number of input/output channels. Given an input topology graph \(G\) (\(X\), \(A\)) containing \(N\) nodes/centroids, each node has a feature vector \(\overrightarrow{x_{i}}\) and dimension \(F\). The attention coefficients \(\alpha_{ij}\) between nodes \(i\) and \(j\) is calculated as:  
\[ \begin{array}{rl}\alpha_{ij}&=softmax\left(attention\left(W.\overrightarrow{x_ {i}},\ W.\overrightarrow{x_{j}}\right)\right)\\ &=softmax(LeakyReLU(\overrightarrow{a}[W.\overrightarrow{x_{i}}||\ W. \overrightarrow{x_{j}}]))\end{array}  \tag{3}\]  
where _attention_() represents a single-layer feedforward neural network with a weight vector \(\overrightarrow{a}\); \(W\) represents the weight matrix for transforming the input features into higher-level features; \(\overrightarrow{r}\) and \(||\) represent the transposition and concatenation operations, respectively; LeakyReLU is a non-linear activation function; and _softmax_() is used to normalize the neighbors of node \(i\) between 0 and 1. After that, GAT aggregates each neighboring node and obtains the embedded features as follows:  
\[\overrightarrow{x_{i}}=\sigma\left(\sum_{j\in N_{i}}\alpha_{ij}\cdot W. \overrightarrow{x_{j}}\right)\overrightarrow{x_{i}}\ \in\ N_{i} \tag{4}\]  
where \(\overrightarrow{x_{i}}\) and \(\overrightarrow{x_{i}}\) represent the input data and embedded features, respectively; \(\sigma\)-_sigmoid_() is used to normalize the output between 0 and 1; \(N_{i}\) denotes the neighbors of node \(i\). From the geological point of view, the geochemical signatures of mineralization-favored spaces inherited from multiple geochemical processes are often anisotropic [7], and Formula (4) indicates that the anisotropy of the node \(i\) in terms of geochemical signatures can be characterized by the relationship weights \(\alpha_{ij}\) of node \(i\)'s neighborhood points \(\{x_{j},j=1,\ 2,\ \ldots\,\ N_{i}\}\), including itself, see Figure 6b. Authors in [34] further pointed out that: in GAT, every node attends to its neighbors given its own representation as the query, but for any query, the neighbor-scoring is monotonic with respect to per-node scores. This could be problematic in real-world applications. To remove this limitation, they proposed GATv2 by just modifying the order of operations in GAT. Their experiments demonstrated that GATv2 outperforms GAT in all benchmarks while having the same parametric cost. Inasmuch, as shown in Figure 5, this study used two GATv2 layers to construct the encoder of OGE, and then, the input  
Figure 6: Illustration of the GAT layer (Modified after [24]): (**a**) how to determine the attention coefficient, corresponding to the Formula (3); (**b**) aggregating features of the node 2-based attention mechanism, with neighborhood {1,3,4,7}, corresponding to the Formula 4.  
geochemical topology graph is learned and mapped to a low-dimensional vector space Z. Our encoder can seamlessly learn the compositional relationships of eleven geochemical variables (i.e., nodal attributes) and their spatial structural features.