On general models of complex networks with some applications

Barabasi and Albert find that many large networks exhibit a scale-free power-law distribution of vertex degrees. We show that when vertex degrees of large networks follow a scale-free power-law distribution with the exponent gamma ges 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order less than log N. Furthermore, let
ⁿ
k be the number of degree-k vertices. In this paper we prove that
ⁿ
1 must be divisible by the least common multiple of k
^gamma

₁
, k
^gamma

₂
,..., k
^gamma

_l
, where 1 = k1 < k2 <...> kl is the degree sequence of the network. Then we construct a general model of networks of scale-free and obtain some detail properties on scale-free and small-world networks. Our method has the benefit of relying on conditions that are static and easily verified. They are verified by many experimental results of diverse real networks and have comprehensive applications to social, natural and synthetic systems.