Paper: arXiv 2601.04096

Authors: Riley James Bendel

Abstract

We study contagion and systemic risk in sparse financial networks with balance-sheet interactions on a directed random graph. Each institution has homogeneous liabilities and equity, and exposures along outgoing edges are split equally across counterparties. A linear fraction of institutions have zero out-degree in sparse digraphs; we adopt an external-liability convention that makes the exposure mapping well-defined without altering propagation. We isolate a single-hit transmission mechanism and encode it by a sender-truncated subgraph G_sh. We define adversarial and random systemic events with shock size k_n = c log n and systemic fraction epsilon n. In the subcritical regime rho_out < 1, we prove that maximal forward reachability in G_sh is O(log n) with high probability, yielding O((log n)^2) cascades from shocks of size k_n. For random shocks, we give an explicit fan-in accumulation bound, showing that multi-hit defaults are negligible with high probability when the explored default set is polylogarithmic. In the supercritical regime, we give an exact distributional representation of G_sh as an i.i.d.-outdegree random digraph with uniform destinations, placing it within the scope of the strong-giant/bow-tie theorem of Penrose (2014). We derive the resulting implication for random-shock systemic events. Finally, we explain why sharp-threshold machinery does not directly apply: systemic-event properties need not be monotone in the edge set because adding outgoing edges reduces per-edge exposure.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs establishing sharp threshold transitions and branching process bounds for financial contagion in sparse random digraphs. However, it is purely theoretical with zero empirical backtesting or real-world financial network data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze contagion & systemic risk in sparse financial networks] --> B[Model Setup: Directed random graph with homogeneous balance sheets & zero out-degree convention]
    B --> C[Single-Hit Mechanism: Construct sender-truncated subgraph G_sh & shock size k_n = c log n]
    C --> D[Subcritical Regime Analysis: rho_out < 1]
    C --> E[Supercritical Regime Analysis: rho_out > 1]
    
    D --> F[Finding: O(log n) reachability yields small cascades O(log n)^2 & negligible multi-hit defaults]
    E --> G[Finding: Mapped to Penrose's Bow-Tie Theorem for exact systemic risk thresholds]
    
    F --> H[Key Outcome: Non-monotone edge-addition effects prevent simple sharp-threshold machinery]
    G --> H