# Knowledge Graph Extrapolation Network with Transductive Learning for Recommendation
## 4 The Proposed Method
### Embedding Propagation  
The embedding propagation is the knowledge propagation with collaborative signal. As is known to all, knowledge is propagated along the links in the knowledge graph. Thus, the user's long-distance interests are captured by capturing higher-order connection information in KG. Furthermore, the user-item collaborative graph with interactions between users and items, directly contains a collaborative signal, including the relationship between original user and item. So, the collaborative signal is extracted by learning interactions in CG. Concretely speaking, the embedding propagation is composed of two components: collaborative signal extraction and knowledge propagation. First, the collaborative signal is extracted as the initial sets of entities for users and items according to historical users' interaction data with the items. Second, knowledge is propagated layer by layer by aggregating information from each layer's neighborhood. Collaborative signal extraction. In user and item interactions, it is generally assumed that users with similar access behaviors may prefer the similar items, and users may like items that are similar to their preferences. Thus, it makes sense for collaboration signals to be embedded in the feature representations of users and items and participate in propagation. In particular, the user's collaborative signal is represented by items, which the user has already indicated they prefer. A collection of these items with a score of 1 between user and items is named the initial set of users for the user. The initial set of users \(u\) formulated as \(\mathcal{I}_{(u)}=\{v|y_{uv}=1\}\). For the item, if different items are all associated with a user, there may be a correlation and similarity between these items. Therefore, first, users who have interacted with the item \(v\) are found, then other items that interact with the same users are treated as collaborative neighbors. The set of item's collaborative neighbors is called the initial set of items, as \(\mathcal{I}_{(v)}=\{v_{u}|u\in\{u|y_{uv}=1\}\wedge\ y_{uv_{u}=1\}\}\). For example, in Figure 1, user \(u1\) prefers items \(v1\) and \(v2\), so the initial set of the user includes \(v1\) and \(v2\). For item \(v1\), the user interacting with \(v1\) is \(u1\) and \(u2\), \(v2\) interacts with \(u1\) and \(v3\) interacts with \(u2\). Thus, the initial set of the \(v1\) contains \(v2\) and \(v3\). Figure 2: The illustration of the proposed KGET framework for recommendation. Notice that the items in the user-item collaborative graph is aligned with the entities in the knowledge graph, to obtain the initial entity set of the user and the initial entity set of the item, respectively. The initial entity set of the user is shown below:  
\[\mathcal{N}^{0}_{(u)}=\{e|(e,v)\in\mathcal{A}\wedge\ v\in\mathcal{I}_{(u)}\}. \tag{1}\]  
The initial entity set of the item is shown below:  
\[\mathcal{N}^{0}_{(v)}=\{e|(e,v_{u})\in\mathcal{A}\wedge\ v_{u}\in\mathcal{I}_{( v)}\} \tag{2}\]  
Knowledge propagation. The knowledge graph contains a large amount of side information, such as item attributes, correlation information and so on. In knowledge graph, entities are connected through relationships. The neighbor information of an entity has an essential influence on the feature of the entity. So, the feature representation of the entity is extended layer by layer through neighborhood knowledge along links between entities themselves. This method is able to capture high-order structural information. Since the initial entity sets of users and items are subsets of the entity set \(\mathcal{E}\) in the knowledge graph, the feature representations of users and items are extended and learned, which starts with them initial entity sets. In other words, both users and items carry out embedding propagation and feature extraction in the form of entities on the knowledge graph. Thus, the propagation of entity \(e\) is used to illustrate the specific propagation process for convenience.