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

TL;DR

This work reframes global BoP imbalances as a directed, weighted network whose stability is governed by spectral properties of the exposure operator. It introduces a nonnegative exposure lift, a BoP adjacency operator, a BoP Laplacian, and a percolation-based phase-transition view to quantify systemic risk and diffusion of shocks. Central contributions include a Spectral Stability Criterion (ρ(A) < 1), a Marginal Spectral Impact framework (edge and node elasticities), and a systemic-risk index (SIRI) combining topology with imbalance magnitudes. The paper also proposes a spectral-clearing policy paradigm—a Global Clearing Union with verifiable spectral safeguards and network-tariff tools—grounded in Collatz–Wielandt certificates and non-backtracking percolation thresholds. Collectively, this framework provides a measurable, topology-driven approach to macroprudential policy that targets network-wide stability rather than isolated country metrics.

Abstract

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 \textbf{Spectral Stability Criterion} proving that the system is globally asymptotically stable if and only if the spectral radius $ρ(A) < 1$, and (iii) a \textbf{Spectral Stability Margin} ($δ= 1 - ρ(B)$) 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.

Figures (2)

• Figure 1: Critical Slowing Down: The relationship between the Spectral Radius $\rho(A)$ and system recovery time $\tau$. As $\rho(A) \to 1$, the system loses its dissipative capacity, leading to infinite shock persistence.
• Figure 2: The Hashimoto non-backtracking constraint. In Section 6.4, the operator $B_{nb}$ acts on directed edges; a transition from $e_1 (i \to j)$ to $e_2 (j \to k)$ is permitted, but the immediate reversal back to $i$ is strictly excluded to isolate systemic circulation from bilateral noise.

Theorems & Definitions (17)

• Remark 2.1: Bilateral netting versus general signed networks
• Proposition 3.1: Spectral stability and finite amplification
• Proposition 4.1: Power iteration and centrality concentration
• proof : Proof sketch
• Remark 4.1: Economic meaning of $r$ in an exposure graph
• Remark 4.2: Economic interpretation of teleportation
• Remark 4.3: Optional refinement: receptivity $\times$ localization
• Remark 4.4: Interpretation: topology as an externality
• Lemma 5.1: Exact Dirichlet-form identity
• proof