We undertake a fundamental study of network equilibria modeled as solutions of fixed point of monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations and is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness encompassing and generalizing several results in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross a certain critical region consisting of the union of finitely many linear submanifolds of co-dimension 1. This is of particular interest in the financial systems context, as it shows that even small shocks affecting the values of the assets of few nodes, can trigger catastrophic aggregated loss to the system and cause the default of several agents.
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