Relationship between degree–rank distributions and degree distributions of complex networks

Both the degree distribution and the degree–rank distribution, which is a relationship function between the degree and the rank of a vertex in the degree sequence obtained from sorting all vertices in decreasing order of degree, are important statistical properties to characterize complex networks. We derive an exact mathematical relationship between degree–rank distributions and degree distributions of complex networks. That is, for arbitrary complex networks, the degree–rank distribution can be derived from the degree distribution, and the reverse is true. Using the mathematical relationship, we study the degree–rank distributions of scale-free networks and exponential networks. We demonstrate that the degree–rank distributions of scale-free networks follow a power law only if scaling exponent λ>2. We also demonstrate that the degree–rank distributions of exponential networks follow a logarithmic law. The simulation results in the BA model and the exponential BA model verify our results.