Financial Contagion Networks as Annealing-Ready Ising Systems: Cascades, Bailout Optimization, and Susceptibility

INTRODUCTION

Modern financial systems are highly interconnected through equity ownership, debt obligations, interbank lending, and derivative contracts. While these interconnections improve liquidity and capital allocation, they also create channels through which financial distress can propagate — the failure of a single institution can trigger a cascade that amplifies a localized shock into a systemic crisis, as demonstrated by the 2008 global financial crisis. Starting from the Elliott–Golub–Jackson (EGJ) financial network model — which describes institutions as interconnected through cross-holdings while owning primitive assets valued outside the network — we extend equilibrium valuation to incorporate solvency thresholds. Once institutions can fail below a critical market value, the linear valuation problem becomes a nonlinear fixed-point system capable of generating cascading failures, motivating a QUBO/Ising-based optimization approach for both identifying worst-case cascades and designing bailout interventions.

METHODOLOGY

We develop the framework in four stages: 1. THRESHOLD-EXTENDED VALUATION: Extending the EGJ equilibrium v = ADp (where A = Ĉ(I−C)⁻¹ is the dependency matrix built from cross-holdings and outside ownership) with threshold-induced failure penalties, yielding a nonlinear fixed-point contagion model v = A(Dp − b(v)). 2. MAXIMUM CASCADE FAILURE PROBLEM (MCFP): Encoding institution failure as binary variables and the contagion-closure constraint as a QUBO whose ground state is the largest self-consistent cascade — mapped to an Ising Hamiltonian and solved via simulated/simulated-quantum annealing. 3. OPTIMAL BAILOUT ALLOCATION PROBLEM (OBAP): Introducing bailout variables as external stabilizing fields on the same Ising system, with a proven 'bailout pinning' condition guaranteeing rescued institutions survive, and reformulating the resulting bi-level problem as a single joint QUBO over failure and bailout variables under a budget constraint. 4. BAILOUT SUSCEPTIBILITY: Using the fluctuation–dissipation theorem on equilibrium Metropolis Monte Carlo samples to define a response function χij quantifying how stabilizing institution j reduces the failure tendency of institution i, yielding an aggregate susceptibility μj and a scalable susceptibility-driven greedy intervention strategy with a (1−1/e) approximation guarantee under submodularity.

RESULTS

Equilibrium valuation on a 20-institution, 5-asset network shows threshold effects can trigger endogenous failure (institution 8 fails, ≈12.5% equilibrium loss) even under modest shocks. Solving the MCFP on a 1000-institution network via simulated quantum annealing (cross-validated with parallel tempering) identifies a self-consistent worst-case cascade of 561 institutions — all 10 mega-banks, 89 of 90 regional institutions, and 462 of 900 peripheral institutions — at minimum QUBO energy −498.50. The joint OBAP QUBO was validated on networks from 20 to 1000 institutions: for the 100-institution hierarchical network, optimal bailout intervention reduces magnetization from m = +0.840 to m = −0.060, with only 40 institutions directly rescued indirectly stabilizing a further 47 (only 9 remain failed of the original 92), demonstrating that targeted bailouts build 'firebreaks' that protect far more institutions than are directly rescued. The bailout susceptibility matrix χ, computed from equilibrium Monte Carlo sampling, shows a pronounced response block among core institutions and weak response among peripheral ones — confirming that interventions on highly interconnected institutions produce disproportionate system-wide stabilization, and providing the physical justification for the susceptibility-driven greedy strategy.

IMPLICATIONS

This work establishes a unified Ising–QUBO framework spanning equilibrium valuation, worst-case cascade analysis, and optimal regulatory intervention, compatible with classical simulated annealing, quantum-inspired optimizers, and quantum annealing hardware. The bailout susceptibility introduced here provides a genuinely dynamical measure of systemic importance — incorporating both network connectivity and collective contagion dynamics — in contrast to purely topological centrality measures. Beyond the specific optimization problems solved, mapping financial contagion onto an interacting spin system opens the door to applying the broader toolkit of statistical mechanics to systemic risk. Extensions under consideration include heterogeneous recovery rates, stochastic asset dynamics, multilayer financial networks, empirical ownership data, and hybrid classical–quantum optimization as annealing hardware matures — offering a route toward scalable stress testing and real-time cascade prediction.

REFERENCES

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