标题： 高阶相互作用的IntComplex 显示英文标题

标题： IntComplex for high-order interactions

摘要： 图可用于建模生物、材料科学和社会网络等不同领域中的成对交互作用。 然而，它们本质上忽略了涉及多于两个实体的交互作用。 单纯复形和超图已成为建模多体交互作用的重要框架；尽管如此，它们在捕捉特定高阶交互作用方面存在局限性，尤其是涉及从$n$交互作用到$m$交互作用的转换。 为解决这一差距，我们提出了 IntComplex 作为一种创新框架，以全面表征此类高阶交互作用。 我们的框架利用同调理论，提供对这种交互作用中固有拓扑结构的定量表示。 IntComplex 被定义为一组交互作用，每种交互作用都可以通过二叉树等效表示。 受 GLMY 同调的启发，我们引入了同调以详细分析跨相邻维度的交互作用形成的结构模式， $p$层同调用于阐明特定维度内$p$交互作用中的环状结构，以及多层同调用于分析跨多个维度的交互作用的环状结构。 此外，我们通过过滤过程引入持久同调，并建立其稳定性，以确保对这些复杂交互作用的稳健定量分析。 提出的 IntComplex 框架为高阶交互作用的拓扑性质分析建立了一个基础范式，显示出显著的潜力，能够推动复杂网络分析领域的进一步发展。 显示英文摘要

摘要： Graphs serve as powerful tools for modeling pairwise interactions in diverse fields such as biology, material science, and social networks. However, they inherently overlook interactions involving more than two entities. Simplicial complexes and hypergraphs have emerged as prominent frameworks for modeling many-body interactions; nevertheless, they exhibit limitations in capturing specific high-order interactions, particularly those involving transitions from $n$-interactions to $m$-interactions. Addressing this gap, we propose IntComplex as an innovative framework to characterize such high-order interactions comprehensively. Our framework leverages homology theory to provide a quantitative representation of the topological structure inherent in such interactions. IntComplex is defined as a collection of interactions, each of which can be equivalently represented by a binary tree. Drawing inspiration from GLMY homology, we introduce homology for the detailed analysis of structural patterns formed by interactions across adjacent dimensions, $p$-layer homology to elucidate loop structures within $p$-interactions in specific dimensions, and multilayer homology to analyze loop structures of interactions across multiple dimensions. Furthermore, we introduce persistent homology through a filtration process and establish its stability to ensure robust quantitative analysis of these complex interactions. The proposed IntComplex framework establishes a foundational paradigm for the analysis of topological properties in high-order interactions, presenting significant potential to drive forward the advancements in the domain of complex network analysis.