# Process Discovery for Event Logs with Multi-Occurrence Event Types
## 2 Related Works
### Link Prediction with GNN  
We selected the GNN approach for the equivalence link prediction, as the GNN has superior efficiency in the standard link-prediction problems. The input feature vectors for the link prediction are generated as low-dimensional embedding vectors describing the context and feature parameters of the nodes and edges. In the GNN, for each node, the engine collects the context description vectors found in the neighboring nodes. The usual approach is to perform random walks and to collect the context vectors of the nodes touched in the walk. In the next phase, the collected context vectors are aggregated into an updated feature vector of the current node. The generated status vectors can be used as input feature vectors of the graph components in the required classification problems. Regarding the concrete embedding generation approaches, we selected the following popular benchmark methods: Node2 Vec, Hin2 Vec, GraphSage, and HinSage. The simplest approach is the Node2 Vec [54] method, which can be applied for homogeneous graphs. In this case, there is only one type of node and edge. The method uses a biased random walk to collect the local status vectors from the neighborhood nodes. The tested node sequences are processed with Skip-Gram model, which learns embeddings by maximizing the likelihood of co-occurring nodes within a defined context window. The simplicity and scalability of Node2 Vec make it effective for homogeneous graphs, but it is limited in its ability to capture semantic relationships in heterogeneous information networks. The Hin2 Vec [55] approach is the extension of the Node2 Vec framework for heterogeneous graphs. In heterogeneous graphs, nodes and edges are associated types; different nodes or edges may belong to different types. This type of information can be used to make constraints on the generated walks during the embedding calculations. Like Node2 Vec, Hin2 Vec uses the Skip-Gram model to learn embeddings, but it outputs not only node embeddings but also link embeddings. GraphSage [56] is a general inductive framework that uses node feature data to generate the embedding vector. The resulting embedding is calculated by sampling and aggregating features from a node's local neighborhood. The proposed method supports both supervised and unsupervised learning; it can be used for node classification/regression, and link prediction for homogeneous networks. The pseudo code of embedding generation is shown in Algorithm 1. ```
Input: Graph: G (V,E); input features: {x\({}_{v}\)}; depth: K; weight matrices: W\({}^{k}\);non-linearity factor: s; differentiable aggregator functions: AGGREGATE\({}_{k}\); neighborhood function: N Output: Vector representation z\({}_{v}\)
1. h\({}^{0}\)\({}_{v}\) = x\({}_{v}\) (for all v in V)
2. for k in 1...K do
3. for v in V do:
4. \(\mathrm{h}^{k}\)N\({}_{(v)}\) = AGGREGATE\({}_{k}\) (\(\mathrm{h}^{k-1}\)\({}_{u}\), for all u in N(v)))
5. \(\mathrm{H}^{k}\)\({}_{v}\) = s(W\({}^{k}\) concat(h\({}^{k-1}\)\({}_{v}\)\(\mathrm{h}^{k}\)N\({}_{(v)}\) ))
6. end
7. \(\mathrm{h}^{k}\)\({}_{v}\) = h\({}^{k}\)\({}_{v}\)/ \(\mid\) \(\mathrm{h}^{k}\)\({}_{v}\)\(\mid\)\({}_{2}\) for all v in V
8. end
9. \(\mathrm{z}_{v}\) = h\({}^{K}\)\({}_{v}\), for all v in V
```  
**Algorithm 1**GraphSAGE embedding generation algorithm  
The HinSage [57] method is the enhancement of the inductive GraphSAGE model for heterogeneous graph representation learning. The method employs a neural aggregation mechanism to generate embeddings using sampling and aggregation steps. For each node, type-specific neighbors are sampled, and separate aggregations are performed for each type. The aggregated embeddings are then combined to update the node's representation. Unlike Node2 Vec and Hin2 Vec, HinSage does not require manually defined meta-paths, as it learns the relationships directly from the data. Additionally, HinSage supports inductive learning, making it suitable for dynamic or partially observed graphs where new nodes or edges are introduced.