Explanation of Link Predictions on Knowledge Graphs via Levelwise Filtering and Graph Summarization

Abstract

Link Prediction methods aim at predicting missing facts in Knowledge Graphs (KGs) as they are inherently incomplete. Several methods rely on Knowledge Graph Embeddings, which are numerical representations of elements in the Knowledge Graph. Embeddings are effective and scalable for large KGs; however, they lack explainability.Kelpie is a recent and versatile framework that provides post-hoc explanations for predictions based on embeddings by revealing the facts that enabled them. Problems have been recognized, however, with filtering potential explanations and dealing with an overload of candidates. We aim at enhancing Kelpie by targeting three goals: reducing the number of candidates, producing explanations at different levels of detail, and improving the effectiveness of the explanations. To accomplish them, we adopt a semantic similarity measure to enhance the filtering of potential explanations, and we focus on a condensed representation of the search space in the form of a quotient graph based on entity types. Three quotient formulations of different granularity are considered to reduce the risk of losing valuable information. We conduct a quantitative and qualitative experimental evaluation of the proposed solutions, using Kelpie as a baseline.

Appendices

A Appendix: Repair of Inconsistent and Unsatisfiable Ontologies

We integrated the datasets DB100K and DB50K with OWL schema axioms retrieved from various sources. The resulting ontologies were inconsistent; hence, we manually repaired them. Moreover, YAGO4-20 turned out to contain unsatisfiable classes. Specifically, we identified the causes of such problems by running the explanation facility of the reasoner. In this appendix, we report some insights on the adjustments that we performed to make DB100K consistent and YAGO4-20 satisfiable, hence “reasonable”. In our GitHub repository, we report all the adaptations.

Table 5. Hyper-parameters of the models

For DB100K, we modified the types for certain instances. For instance, our SPARQL query to retrieve the class assertions returned $\textit{Politician}$, and $\textit{TimePeriod}$ for several entities. Such types led to inconsistencies as $\textit{Politician}$ is a subclass of $\textit{Person}$ which, in turn, is disjoint with $\textit{TimePeriod}$. We modified these entities, keeping only the type $\textit{Politician}$.

We also modified schema axioms in certain cases to preserve several triples, which otherwise would have led to inconsistencies. For instance, we modified the range of the property $\textit{location}$. We recall that the range of a relationship p specify the classes whose instances can occur as object in a triple with predicate p. We changed the range from $\textit{Place}$ to $\textit{Place} \sqcup \textit{Company}$. Through this adjustment, we accommodate also triples having $\textit{location}$ as predicate and an instance of $\textit{Company}$ as object.

In other cases, we needed to remove triples. For instance, we removed the triple $\langle \textit{Subramanian\_Swamy}, \textit{region}, \textit{Economics} \rangle$. These triples caused an inconsistency because the type of $\textit{Economics}$ is, $\textit{University}$ which in turn is a descendant of $\textit{Agent}$ in the class hierarchy. However, the range of $\textit{region}$ is $\textit{Place}$ which is disjoint with $\textit{Agent}$.

For YAGO4-20, we removed certain $\textit{subClassOf}$ axioms. For instance, the class $\textit{Districts\_of\_Slovakia}$ was a subclass of $\textit{AdministrativeArea}$, $\textit{Product}$, and $\textit{CreativeWork}$. The class $\textit{Districts\_of\_Slovakia}$ was unsatisfiable as $\textit{AdministrativeArea}$ is subclass of $\textit{Localization}$ which is disjoint with $\textit{CreativeWork}$. We modified it keeping only $\textit{AdministrativeArea}$ and $\textit{Product}$ as super-classes.

B Appendix: Hyper-parameters

In this appendix, we report in Table 5 the hyper-parameters that we adopted to train each model on each dataset. Furthermore, we employed the same set of hyper-parameters to execute the post-training in the extraction of the explanations and to retrain the models in the evaluation of the explanations.

Note that:

• D is the embedding dimension, in the models that we adopted entity and relation embeddings always have same dimension

• p is the exponent of the p-norm

• Lr is the learning rate

• B is the batch size

• Ep is the number of epochs

• $\gamma$ is the margin in the Pairwise Ranking Loss

• N is the number of negative triples generated for each positive triple

• $\omega$ is the size of the convolutional kernels

• Drop is the training dropout rate, specifically:

  • in is the input dropout

  • h is the dropout applied after a hidden layer

  • feat is the feature dropout

We adopted random search to find the values of the hyper-parameters. Exceptions are given by B and Ep. For B we adopted the value leading to optimize execution times and parallelism. For Ep we adopted early stopping with 1000 as maximum number of epochs and 5 as patience threshold during the training of the models, and we reported the epoch on which the training stopped. Hence, we used such value as number of epochs in the post-training and in the evaluation. Furthermore, as in Kelpie, for TransE we adopted the learning rate (Lr) values in Table 5 during training and evaluation, but for the post-training we used a different value. For TransE the batch size (B) is particularly large (2048) and usually exceeds by far the number of triples featuring an entity. This affects post-training because in any post-training epoch the entity would only benefit from one optimization step. We easily balanced this by increasing the Lr to 0.01.