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.
Financial Contagion Networks as Annealing-Ready Ising Systems Cascades, Bailout Optimization, and Susceptibility
The Institute of Mathematical Sciences (IMSc)（数学科学研究所）；QCAR Group, The Institute of Mathematical Sciences (IMSc)（量子计算与算法研究组，数学科学研究所）；Pecslab Research（Pecslab 研究）；Department of Physics, Ramjas College, University of Delhi（德里大学拉姆贾斯学院物理系）；Homi Bhabha National Institute（霍米·巴巴国家研究所）
