Abstract
Many real-world networks exhibit scale-free degree distributions together with dissortative degree correlations. Such networks exhibit interesting structural and dynamical features, especially in the limit of maximal correlations. In the latter case, the network vertices are shown to form nested bilayers, the number of which grows with network size N but saturates for large N. This bilayer structure strongly affects the properties of dynamical processes on such networks and implies a large number of attractors that govern the long-time behavior of these processes. Surprisingly, the most complex dynamical behavior is found for intermediate rather than for large network sizes.