Uniform transformation and collective degree analysis on higher-order networks

Hypergraphs provide crucial and potent mathematical models for accurately describing the intricate high-order interactions prevalent in real-world systems. To advance the research landscape of hypergraph theory and deepen its applications, a systematic investigation into the group properties of high-order networks that model these systems is imperative. In this context, we introduce an innovative method for transforming general non-uniform hypergraphs into uniform hypergraphs, grounded in hypergraph theory, set theory, and statistical mechanics. This approach aims to uncover the complex group organization of the corresponding systems, significantly preserving linear operations, and thereby mitigating the complexity commonly associated with tensor-based hypergraph computations. The refined concepts and analytical tools we have developed are crucial for assessing the distribution and importance of groups of varying sizes. For each of these two practical challenges, we have conducted experiments using two different real-world datasets. Our research findings have substantially advanced hypergraph theory, while also providing valuable insights for analyzing group characteristics in higher-order networks based on hypergraphs, thereby expanding the application scope of network science.