The Heat Kernel Expansion: Scalar Curvature for Shock Detection in Higher-Order Financial Networks

Abstract

This work follows the evolution of financial networks in Norway over a period of nine years at a monthly rate. The data consists of board directors and their affiliations to companies, which we model as simplicial complexes. In this framework, directors are represented as nodes and companies as faces of the com plex. To characterize the latter, we focus on two measures: one topological (the Euler characteristic computed through the Betti numbers), and another geometrical (curvature). The latter is computed from the coefficients of the series expansion of the heat kernel in powers of time, which is our major contribution in this work. The change in both measures reveals variation in the board interlock due to legisla tion, and serves as a measure for detecting shocks in networks. Inflection points in curvature are associated with external forcing, and minima with shock arrival times.