Vertex-degree sequences in complex networks: New characteristics and applications

Many complex networks exhibit a scale-free vertex-degree distribution in a power-law form c k − γ , where k is the vertex-degree variable and c and γ are constants. To better understand the mechanism of power-law formation in real-world networks, it is effective to explore 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 k 1 < k 2 < ⋯ < k l , where { k 1 , k 2 , … , k l } 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 log N . In the present paper, we further study complex networks with an exponential vertex-degree distribution and prove that the same conclusion also holds. In addition, we verify our claim by showing many real-world examples. We finally discuss some applications of the new finding in various fields of science and technology.