Trading networks and Hodge theory*

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'.