The Characteristic and Verification of Length of Vertex-Degree Sequence in Scale-Free Network

Many natural large-sized complex networks exhibit a scale-free, power-law distribution of vertex degree. To better understand the formation mechanism of power law in the real network, we analyze the general nature in scale-free network based on the vertex-degree sequence. We show that when the power exponent of scale-free network is greater than 1, the number of degree-k 1 vertices, when nonzero, is divisible by the least common multiple of 1, k 2 γ /k 1 γ , …, k i γ /k 1 γ , and the length of vertex-degree sequence l is of order log N, where 1 ≤ k 1 < k 2 < … < k l is the vertex-degree sequence of the network and N is the size of the network. We verify the conclusion by the coauthorship network DBLP and many other real networks in diverse domains.