The ability of complex systems to be stable has, for good reasons, been widely debated and studied. In this work we try to understand why some mathematical models of complex systems suggest that their probability of stability is very low. We will show that it is the full structure of the system that defines its stability, and it therefore cannot be deduced from overly simplified mathematical models. In particular the complexity of the network cannot be linked to its stability without considering other relevant factors. We then go on to look at how such stable complex networks can evolve and show that minimal input can be sufficient to grow networks that are almost always stable.

The rest of our work focusses on a different type of robustness, the robustness to shock propagation. In particular, we are interested in the impact of uncertainty on optimised networks systems and the methods that can be used to reduce it. In order to do so we consider a lattice model and a simple network model of the financial system and look at the fragility that arises from their optimisation under uncertainty. We find that using a more limited amount of information during the optimisation process, and reducing the level of optimisation can help limit the fragility of the systems obtained. Finally, we consider whether simple measures can help predict the riskiness of the financial system under uncertainty. We use three measures that are not only simple but meaningful for the system considered, and we find that more often than not such measures do better than the usual, more complex methods.