Beyond-triple fact representations including hyper-relational facts with auxiliary key-value pairs, temporal facts with additional timestamps, and nested facts implying relationships between facts, are gaining significant attention. However, existing link prediction models are usually designed for one specific type of facts, making it difficult to generalize to other fact representations. To overcome this limitation, we propose a Unified Hierarchical Representation learning framework (UniHR) for unified knowledge graph link prediction. It consists of a unified Hierarchical Data Representation (HiDR) module and a unified Hierarchical Structure Learning (HiSL) module as graph encoder. The HiDR module unifies hyper-relational KGs, temporal KGs, and nested factual KGs into triple-based representations. Then HiSL incorporates intra-fact and inter-fact message passing, focusing on enhancing the semantic information within individual facts and enriching the structural information between facts. Experimental results across 7 datasets from 3 types of KGs demonstrate that our UniHR outperforms baselines designed for one specific kind of KG, indicating strong generalization capability of HiDR form and the effectiveness of HiSL module. Code and data are available at https://github.com/Lza12a/UniHR.

Reasoning with knowledge graphs (KGs) has primarily focused on triple-shaped facts. Recent advancements have been explored to enhance the semantics of these facts by incorporating more potent representations, such as hyper-relational facts. However, these approaches are limited to \emph{atomic facts}, which describe a single piece of information. This paper extends beyond \emph{atomic facts} and delves into \emph{nested facts}, represented by quoted triples where subjects and objects are triples themselves (e.g., ((\emph{BarackObama}, \emph{holds\_position}, \emph{President}), \emph{succeed\_by}, (\emph{DonaldTrump}, \emph{holds\_position}, \emph{President}))). These nested facts enable the expression of complex semantics like \emph{situations} over time and \emph{logical patterns} over entities and relations. In response, we introduce NestE, a novel KG embedding approach that captures the semantics of both atomic and nested factual knowledge. NestE represents each atomic fact as a $1\times3$ matrix, and each nested relation is modeled as a $3\times3$ matrix that rotates the $1\times3$ atomic fact matrix through matrix multiplication. Each element of the matrix is represented as a complex number in the generalized 4D hypercomplex space, including (spherical) quaternions, hyperbolic quaternions, and split-quaternions. Through thorough analysis, we demonstrate the embedding's efficacy in capturing diverse logical patterns over nested facts, surpassing the confines of first-order logic-like expressions. Our experimental results showcase NestE's significant performance gains over current baselines in triple prediction and conditional link prediction. The code and pre-trained models are open available at https://github.com/xiongbo010/NestE.