In this paper, a new computational topological framework for hypergraph analysis and recognition is developed. “Topology provides scale” is the principle at the core of this set of algebraic topological tools, whose fundamental notion is that of a scale-space topological model (-model). The scale of this parameterized sequence of algebraic hypergraphs, all having the same Euler-Poincaré characteristic than the original hypergraph , is provided by its relational topology in terms of evolution of incidence or adjacency connectivity maps. Its algebraic homological counterpart is again an -model, allowing the computation of new topological characteristics of , which far exceeds current homological analytical techniques. Both scale-space algebraic dynamical systems are hypergraph isomorphic invariants. The hypergraph isomorphism problem is attacked here to demonstrate the power of the proposed framework, by proving the ability of -models to differentiate challenging cases that are difficult or even infeasible for state-of-the-art practical polynomial solvers. The processing, analysis, classification and learning power of the -model, at both combinatorial and algebraic levels, augurs positive prospects with respect to its application to physical, biological and social network analysis.

中文翻译：

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超图的拓扑尺度框架


本文开发了一种用于超图分析和识别的新计算拓扑框架。 “拓扑提供尺度”是这套代数拓扑工具的核心原理，其基本概念是尺度空间拓扑模型（-model）。这个参数化的代数超图序列的尺度，都具有与原始超图相同的欧拉-庞加莱特征，由其在关联或邻接连通图的演化方面的关系拓扑提供。它的代数同调对应物又是一个 模型，允许计算 的新拓扑特征，这远远超出了当前的同调分析技术。两个尺度空间代数动力系统都是超图同构不变量。这里攻击超图同构问题是为了证明所提出的框架的强大功能，通过证明模型区分具有挑战性的情况的能力，这些情况对于最先进的实用多项式求解器来说是困难的甚至是不可行的。该模型在组合和代数层面上的处理、分析、分类和学习能力预示着其在物理、生物和社会网络分析中的应用前景积极。