This paper investigates systemic risk measures in stochastic financial networks of explicitly modelled bilateral liabilities. We extend the notion of systemic risk from Biagini, Fouque, Fritelli and Meyer-Brandis (2019) to graph structured data. This means that the systemic risk of a stochastic financial network is defined as the minimal amount of bailout capital needed to make the aggregated loss of the system acceptable in the sense of some univariate risk measure. One suitable aggregation function can be derived from a market clearing algorithm as proposed by Eisenberg and Noe (2001). In this setting we show the existence of optimal random bailout capital allocations that distribute the minimal bailout capital and save the network. Further, we study numerical methods for the approximation of systemic risk and optimal allocations of the bailout capital. We propose to use graph neural networks (GNNs) for computing approximately optimal bailout capital and compare their performance to several benchmark allocations. One feature of GNNs is that they respect permutation equivariance of the underlying graph data. In numerical experiments we find evidence that methods that respect permutation equivariance are superior to other approaches.