# An Inference Framework of Markov Logic Network for Link Prediction in Heterogeneous Networks
## 2 Related Work  
In this section, we provide a comprehensive review of existing research on LP in both homogeneous and heterogeneous networks, along with an examination of inferences within MLN. **LP on homogeneous network. ** Heuristic and learning-based methods represent recent trends in LP research on homogeneous networks. Heuristic methods, such as utilizing common neighbors [14; 15], are employed to infer link existence. Recent advancements in LP research have shown that learning-based methods like matrix factorization and network embedding techniques are more effective in acquiring node embeddings [16]. For instance, MHGCN+ [17] and HL-GNN [18] aggregate node embeddings through heterogeneous meta-path interaction and intra-layer propagation with inter-layer connections, respectively, to learn node embeddings. Moreover, ensemble methods [9; 19] enhance LP tasks by combining the outcomes of heuristic and learning-based methods, leading to improved robustness at the expense of efficiency. These approaches construct classifier features through network embedding but struggle to achieve a unified representation of multiplex edges, prevalent in real-world heterogeneous network scenarios. **LP on heterogeneous network. ** Deep neural network-based methods have garnered significant attention for their adept feature extraction in LP tasks on heterogeneous networks [20; 21]. These methods primarily focus on mining type-specific features to optimize prediction performance and subsequently combine relation embeddings through algebraic or transferable operations, as discussed in studies such as [7; 8]. However, effectively capturing inter-dependencies among different types of links remains a challenge. Various techniques have been suggested to create link prediction models that minimize the loss associated with the representation of multi-type links. For instance, HRMNN [22] integrates a relational graph generator that leverages the topological attributes of heterogeneous graphs and combines object-level aggregation with a multi-head attention mechanism to produce more comprehensive node representations. MTTM [2] consists of a generative predictor and a discriminative classifier for link representations, enabling the discrimination of links and leveraging an adversarial neural network to maintain robustness against type differences. While these methods improve the representation of multiplex edges, challenges related to limited observed links and complex network architectures persist. Researchers have endeavored to employ probabilistic graphical models (PGMs) to represent multi-type links in a unified manner, as discussed in [23]. First-order logic formulas have been utilized to model multi-relational databases with Bayesian network (BN), as outlined in [24], and the first-order BN serves as a statistical relational model capturing database frequencies. As MLN formulas conclude predicates indicating various link types, inter-dependencies among various link types could be captured by the weights of formulas [25]. Meanwhile, MLN, acting as a PGM incorporating first-order logic rules, has been applied for representing relational data [26]. By converting heterogeneous networks into MLN and leveraging formulas, it enables a unified expression for multiplex edges. Nonetheless, optimization using SRL methods, as discussed in [27], is found to be non-scalable for LP tasks, primarily due to its inefficiency. Furthermore, the extensive structure of MLN leads to a vast search space and necessitates a costly inference process. **Inference in MLN. ** Efforts have been undertaken to enhance the efficiency and scalability of inferences in MLN for LP across diverse scenarios. For instance, QNMLN [25] expands on the formulas, while MCLA [28] refines its parameter updates for a range of practical tasks. The VEM algorithm [29] serves as the inference framework for MLN, with graph neural networks (GNNs) utilized in the E-step to capture the features of target relation nodes.