Degree distribution in discrete case

Vertex degree of many network models and real-life networks is limited to non-negative integer. By means of measure and integral, the relation of the degree distribution and the cumulative degree distribution in discrete case is analyzed. The degree distribution, obtained by the differential of its cumulative, is only suitable for continuous case or discrete case with constant degree change. When degree change is not a constant but proportional to degree itself, power-law degree distribution and its cumulative have the same exponent and the mean value is finite for power-law exponent greater than 1.

► Degree change is the crux for using the cumulative degree distribution method.
► It suits for discrete case with constant degree change.
► If degree change is proportional to degree, power-law degree distribution and its cumulative have the same exponent.
► In addition, the mean value is finite for power-law exponent greater than 1.