Before revision:  In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures{\em   .
After revision:  From the perspective of systemic risk it is important to understand the influence of this network structure on default contagion. Using sparse random graphs to model the financial network, asymptotic methods turned out powerful to analytically describe the contagion process and to make statements about resilience. So far, however, they have been limited to so-called{\em .