Computing Systemic Risk Measures with Graph Neural Networks

TL;DR

This work extends systemic risk measures to graph-structured, stochastic financial networks by allowing random asset vectors and random interbank liabilities under the Eisenberg–Noe contagion framework. It develops a reformulation based on inner and outer risks, proves the existence of optimal random bailout allocations, and derives an iterative, sample-based algorithm to approximate the risk measures. To scale computation, the authors introduce permutation-equivariant neural architectures—GNNs, PENNs, and their extended XPENN variant—that respect graph structure and node labeling symmetry, with universal approximation results for permutation-equivariant functions. Numerical experiments on synthetic networks show that permutation-equivariant models, especially XPENN, consistently outperform non-graph baselines in learning optimal bailout allocations and calculating systemic risk measures, highlighting the practical potential for scenario-dependent capital allocations in reducing required bailout capital. The results suggest a promising integration of graph-based deep learning with systemic risk theory to enable scalable, robust risk assessment in interconnected financial systems.

Abstract

This paper investigates systemic risk measures for stochastic financial networks of explicitly modelled bilateral liabilities. We extend the notion of systemic risk measures from Biagini, Fouque, Fritelli and Meyer-Brandis (2019) to graph structured data. In particular, we focus on an aggregation function that is derived from a market clearing algorithm proposed by Eisenberg and Noe (2001). In this setting, we show the existence of an optimal random allocation that distributes the overall minimal bailout capital and secures the network. We study numerical methods for the approximation of systemic risk and optimal random allocations. We propose to use permutation equivariant architectures of neural networks like graph neural networks (GNNs) and a class that we name (extended) permutation equivariant neural networks ((X)PENNs). We compare their performance to several benchmark allocations. The main feature of GNNs and (X)PENNs is that they are permutation equivariant with respect to the underlying graph data. In numerical experiments we find evidence that these permutation equivariant methods are superior to other approaches.

Figures (1)

• Figure 1: Both network types for $N=100, i=1$. On the left, the default cascade starting at $i=1.$ On the right, the star-shaped network with centre $i=1.$

Theorems & Definitions (16)

• proof
• proof : Proof of Proposition \ref{['prop:embedding']}
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof