Measuring degree‐dependent failure in scale‐free networks of bipartite structure

We study degree‐dependent failure in networks consisting of two types of nodes with edges running only between nodes of unlike type. Both types of nodes are assumed to have scale‐free degree distributions. These networks are called scale‐free bipartite graphs, which appear in many real‐life networks. In this article, we measure network robustness and fragility based on a recently proposed spectral measure, natural connectivity, which is an average eigenvalue obtained from the graph spectrum. During the edge removal process, an edge between nodes with degrees k and l is retained with a probability proportional to (kl)^‐α, where α > 0 is a biased exponent, so that the most connected nodes are depreciated first. A clear division between robust regime and fragile regime can be extracted based on natural connectivity by tuning the exponent α. The theory is verified by the numerical simulations.