Degree sequences beyond power laws in complex networks

Many complex networks possess vertex-degree distributions in a power-law form of ck^−γ, where k is the degree variable and c and are con- stants. To better understand the mechanism of power-law formation in real- world networks, it is effective to analyze their degree variable sequences. We had shown before that, for a scale-free network of size N,if its vertex-degree sequence is k₁2 1, where {k₁, k₂,…,k₁} is the set of all unequal 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 logN. In the present paper, we further study complex networks with more general distributions and prove that the same conclusion holds even for non-network type of complex systems. In addition, we support the conclusion by verifying many real-world network and system examples. We finally discuss some potential applications of the new finding in various fields of science, technology and society.