Cascading failures on networks with asymmetric dependence

Networks with mutual dependence have been shown to be much more vulnerable to random failures and targeted attacks than those without. However, in real networks, the dependence between two nodes is not always mutual. Periphery nodes may depend on hub nodes, yet the converse is not necessarily true. Considering this asymmetric dependence, we propose a model of cascading dynamics of networks, where the dependence between nodes is determined by their degrees. We find that the asymmetric dependence makes networks more robust than the symmetric one, and the percolation transition point is not sensitive to the number of the asymmetric dependence nodes. Furthermore, scale-free networks with asymmetric dependence can still be very robust to random failures, rather than extremely fragile as the one with mutual dependence. We also develop an approach to analyse this model and obtain the exact solutions for the size of the giant component and the critical point. Both simulation and analytical results reveal the existence of the crossover between the first- and the second-order percolation transitions in our model.