Biased edge failure in scale-free networks based on natural connectivity

The natural connectivity is recently reported as a novel spectral measure of robustness in complex networks. It has a clear physical meaning and a simple mathematical formulation. In this article, based on the natural connectivity, we study the robustness and fragility of scale-free networks under topologically biased failure (degree-dependent percolation). The probability p (ij) that an edge between nodes i and j remains intact is assumed to depend on the degrees k (i) and k (j) through p (ij) ae (k (i) k (j) )(-alpha). We show that a clear division between robust regime and fragile regime can be extracted based on natural connectivity by tuning the bias exponent alpha. The theory is found in good accordance with simulations.