The Spectral Topology of Global Imbalances:A Graph-Theoretic Framework for Systemic Risk in the Balance of Payments

Traditional balance-of-payments (BoP) analysis treats national external positions as largely idiosyncratic time series. This misses an essential structural fact: global imbalances are jointly realized on a directed, weighted network of cross-border current-account and financial claims. We propose a network-theoretic paradigm in which the world economy is a directed graph whose edge weights encode net bilateral exposures. In this setting, systemic fragility is an emergent property of the spectral topology of the global exposure matrix. We develop (i) a mathematically explicit construction of a BoP adjacency operator, (ii) a $\mathbf{\text{Spectral}}\mathbf{\text{Stability}}\mathbf{\text{Criterion}}$ proving that the system is globally asymptotically stable if and only if the spectral radius $\rho (A)<1$, and (iii) a $\mathbf{\text{Spectral}}\mathbf{\text{Stability}}\mathbf{\text{Margin}}$ ($\delta =1-\rho (B)$) that quantifies the proximity of the global economy to a Critical Slowing Down'' phase transition. Furthermore, we define a systemic-risk index using eigenvector centrality to identify nodes whose failure is mathematically indistinguishable from global collapse. Finally, we employ a \textbf{Non-backtracking (Hashimoto) operator} to derive a precise \textbf{topological threshold} for sovereign debt contagion, filtering bilateral noise’’ to isolate deep-network circulation. Our results demonstrate that systemic risk is a latent property of the global spectral topology, requiring macroprudential interventions targeted at the network’s spectral gaps rather than individual debt-to-GDP ratios.