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We study the mean field approximation of a recent model of cascades onnetworks relevant to the investigation of systemic risk control in financialnetworks. In the model, the hypothesis of a trend reinforcement in thestochastic process describing the fragility of the nodes, induces a trade-offin the systemic risk with respect to the density of the network. Increasing theaverage link density, the network is first less exposed to systemic risk, whileabove an intermediate value the systemic risk increases. This result offers asimple explanation for the emergence of instabilities in financial systems thatget increasingly interwoven. In this paper, we study the dynamics of theprobability density function of the average fragility. This converges to aunique stable distribution which can be computed numerically and can be used toestimate the systemic risk as a function of the parameters of the model.

Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks