We study optimal intervention design problems aimed at minimizing the aggregate volatility in network systems. We consider linear network equilibrium models arising in network games, production networks, and opinion dynamics, where exogenous shocks hitting a few nodes can propagate through the network and cause detrimental systemic effects. Focusing on idiosyncratic shocks, we first identify a general network centrality measure that determines the amplification factor of the variance of each node's shock to the aggregate system volatility, and show that, in relevant special cases, it reduces to Katz-Bonacich, cycle, and concentration centralities. We then consider a system planner who can attenuate the shocks on the nodes, subject to a budget constraint, so as to minimize the aggregate volatility. We solve this problem analytically and show that its solution has a water-filling-type structure: for low budget, the intervention is concentrated on the node(s) maximizing a quantity ω_i that combines the shock's variance, the node's centrality, and the intervention cost; as the budget increases, more and more nodes are targeted, and for sufficiently large budget every node with ω_i > 0 receives an intervention proportional to ω_i. Finally, we study a robust version of the problem, where the planner minimizes the volatility against the worst-case vector of shock variances chosen by an adversary within a given budget. We solve this min-max problem analytically and show that its solution also has a water-filling-type structure, with the ratio between the node's centrality and the square root of the adversarial cost playing the role of ω_i.

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