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  .
We undertake a fundamental study of network equilibria modeled as solutions of fixed point equations for 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. It also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents . In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits .
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It also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities.
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several results
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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
certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses
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. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits