The problem of analyzing interconnectedness is one of today’s premier challenges in understanding systemic risk. Connections can both stabilize networks and provide pathways for contagion. The central problem in such networks is establishing global behavior from local interactions. Jiang-Lim-Yao-Ye (Jiang et al 2011 Mathematical Programming 127 1 203–244) recently introduced the use of the Hodge decomposition (see Lim 2020 SIAM Review 62 685–715 for a review), a fundamental tool from algebraic geometry, to construct global rankings from local interactions (see Barbarossa et al 2018 (2018 IEEE Data Science Workshop (DSW), IEEE) pp 51–5; Haruna and Fujiki 2016 Frontiers in Neural Circuits 10 77; Jia et al 2019 (Proc. of the XXV ACM SIGKDD International Conf. on Knowledge Discovery & Data Mining, pp 761–71 for other applications). We apply this to a study of financial networks, starting from the Eisenberg-Noe (Eisenberg and Noe 2001 Management Science 47 236–249) setup of liabilities and endowments, and construct a network of defaults. We then use Jiang-Lim-Yao-Ye to construct a global ranking from the defaults, which yields one way of quantifying ‘systemic importance’.

分析互连性的问题是当今理解系统性风险的主要挑战之一。连接既可以稳定网络，又可以提供传播途径。这种网络的中心问题是通过局部交互来建立全局行为。Jiang-Lim-Yao-Ye（Jiang et al 2011 Mathematical Programming 127 1 203–244）最近引入了Hodge分解的使用（有关综述， 请参阅Lim 2020 SIAM评论62 685–715），这是代数几何的基本工具，从本地互动中构建全球排名（请参见Barbarossa等人， 2018年（2018年IEEE数据科学研讨会（DSW），IEEE），第5–5页； Haruna和Fujiki，2016年）神经回路前沿 10 77; Jia等， 2019年（XXV ACM SIGKDD国际知识发现和数据挖掘大会，其他应用程序，第761-71页）。我们将其应用于金融网络的研究，从Eisenberg-Noe（Eisenberg and Noe 2001 Management Science 47 236-249）负债和end赋的建立开始，并构建一个违约网络。然后，我们使用Jiang-Lim-Yao-Ye从默认值构建全局排名，这是量化“系统重要性”的一种方法。