Default Contagion, Matrix Approximation, and Control in Sparse Financial Networks

We study systemic default contagion in sparse financial networks and develop a framework for deciding when aggregate exposure matrices are reliable and when node-level network information changes tail risk and control design. The first contribution is a multi-population McKean-Vlasov foundation for distance-to-default dynamics with common noise, bounded state-dependent killing, loss feedback, sparse weighted exposures, and regulatory intervention, including quantitative convergence, propagation of chaos, stability in contagion matrices, controlled well-posedness, a two-population HJB characterization, and a steep-killing bridge to absorbing-boundary contagion. The second contribution is a set of computable matrix-approximation diagnostics: finite-grid bounds driven by row-exposure dispersion and square-edge spread, constructive tail-loss gaps for networks sharing the same aggregate matrix, and a spectral-radius criterion for local cascade onset. The third contribution is an information-value theory for control, showing that node-level graph pressure has strictly positive value when within-type pressure variation interacts with nonsaturated marginal killing reduction. Matched sparse-graph and matrix experiments, common-noise tests, HJB feedback diagnostics, fixed-budget control comparisons, and EBA/Pillar 3-calibrated synthetic networks validate the framework. The main conclusion is that finite-type matrices are effective in regular-mixing regimes, whereas concentrated sparse exposures generate tail-risk and intervention effects that require local-pressure diagnostics and network-aware control.