Relationship between degree-rank function and degree distribution of protein-protein interaction networks

It is argued that both the degree-rank function r=f(d), which describes the relationship between the degree d and the rank r of a degree sequence, and the degree distribution P(k), which describes the probability that a randomly chosen vertex has degree k, are important statistical properties to characterize protein-protein interaction (PPI) networks, both rank-degree plot and frequency-degree plot are reliable tools to analyze PPI networks. An exact mathematical relationship between degree-rank functions and degree distributions of PPI networks is derived. It is demonstrated that a power law degree distribution is equivalent to a power law degree-rank function only if scaling exponent is greater than 2. The puzzle that the degree distributions of some PPI networks follow a power law using frequency-degree plots, whereas the degree sequences do not follow a power law using rank-degree plots is explained using the mathematical relationship.