计算有向图与箭图的奇异单纯同调
Computing singular simplicial homologies of digraphs and quivers
AI总结：
本文为有向图与箭图发展同伦理论，构造函子实现奇异单纯同调与空间同调的同构，并提供计算持久同调的高效算法，适用于大规模网络分析。
摘要：
大型复杂系统的动力学主要通过成对相互作用来建模，其基本结构是形如有向图或箭图的网络。在众多科学学科中，将定向旗复形的同调理论应用于研究作为网络出现的有向图已取得了显著成功。与其他有向图同调相比，这种有向团同调在计算上相对容易，使其在大规模网络应用中更具优势。通过将奇异单纯同调的思想扩展到不同态射类型范畴中的箭图，最近已构建了几种新的奇异单纯同调理论。\n 一般而言，箭图的高效计算同调此前并未被认真考虑。在本文中，我们进一步发展了箭图的同伦理论，以推导出实现奇异单纯箭图同调与某些空间同调之间同构的函子。这些空间的单纯链以紧凑的形式出现，其便利性至少与用于计算的定向旗复形相当。\n 此外，我们的构造在同调同构方面是自然的，使其适用于与持久同调结合的实际应用。特别地，对于所考虑的每种奇异单纯同调，我们提供了计算其持久同调的高效算法。
英文摘要：
The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as networks within numerous scientific disciplines. This homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large networks. By extending the ideas of singular simplicial homology to quivers in categories of different morphism types, several new singular simplicial homology theories have recently been constructed. Computationally efficient homologies for quivers have in general not previously been seriously considered. In this paper we develop further the homotopy theory of quivers necessary to derive functors that realise isomorphisms between the singular simplicial quiver homologies and the homologies of certain spaces. The simplicial chains of these spaces arise in a conveniently compact form that is at least as convenient as the directed flag complex for computations. Moreover, our constructions are natural with respect to the isomorphisms on homology making them suitable for applications in conjunction with persistent homology for practical use. In particular, for each of the singular simplicial homologies considered, we provide efficient algorithms for the computation of their persistent homology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
