  Recent work on knowledge graph completion (KGC) focused on learning
embeddings of entities and relations in knowledge graphs. These embedding
methods require that all test entities are observed at training time, resulting
in a time-consuming retraining process for out-of-knowledge-graph (OOKG)
entities. To address this issue, current inductive knowledge embedding methods
employ graph neural networks (GNNs) to represent unseen entities by aggregating
information of known neighbors. They face three important challenges: (i) data
sparsity, (ii) the presence of complex patterns in knowledge graphs (e.g.,
inter-rule correlations), and (iii) the presence of interactions among rule
mining, rule inference, and embedding. In this paper, we propose a virtual
neighbor network with inter-rule correlations (VNC) that consists of three
stages: (i) rule mining, (ii) rule inference, and (iii) embedding. In the rule
mining process, to identify complex patterns in knowledge graphs, both logic
rules and inter-rule correlations are extracted from knowledge graphs based on
operations over relation embeddings. To reduce data sparsity, virtual neighbors
for OOKG entities are predicted and assigned soft labels by optimizing a
rule-constrained problem. We also devise an iterative framework to capture the
underlying relations between rule learning and embedding learning. In our
experiments, results on both link prediction and triple classification tasks
show that the proposed VNC framework achieves state-of-the-art performance on
four widely-used knowledge graphs. Further analysis reveals that VNC is robust
to the proportion of unseen entities and effectively mitigates data sparsity.
