Characterizing general scale-free networks by vertex-degree sequences

Many complex networks possess a scale-free vertex-degree distribution in a power-law form of ck(-γ), where k is the vertex-degree variable and c and γ are constants. To better understand the mechanism of the power-law formation in scale-free networks, it is important to understand and analyze their vertex-degree sequences. We had shown before that, for a scale-free network of size N, if its vertex-degree sequence is k1 <k2<⋯<kl, where {k1,k2,...,kl} is the set of all non-equal vertex degrees in the network, and if its power exponent satisfies γ>1, then the length l of the vertex-degree sequence is of order log N. In the present paper, we further study complex networks with a more general vertex-degree distribution, not restricted to the power-law, and prove that the same conclusion holds as well. In addition, we verify the new result by real data from a large number of real-world examples. We finally discuss some potential applications of the new finding in various fields of science, technology, and society.