Characterizing vertex-degree sequences in scale-free networks

Many large-scale complex networks exhibit a scale-free vertex-degree distribution in a power-law form. To better understand the mechanism of power-law formation in real-world networks, we explore and analyze the underlying mechanism based on the vertex-degree sequences of such networks. We show that for a scale-free network of size N , if its vertex-degree sequence is k 1 < k 2 < ⋯ < k l , and if its power exponent satisfies γ > 1 , then the length l of the vertex-degree sequence is of order log N . We verify this conclusion by a co-authorship network and some other real networks in various areas.