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.