复杂网络的度分布研究

On Degree Distribution of Complex Network

复杂网络的度分布与其拓扑结构紧密相关。绝大多数复杂网络具有无标度性（Scalef ree），其幂律度分布完全由度分布指数所确定。文中全面研究了复杂网络的度分布指数与其拓扑结构、形成原因以及传播动力学之间的关系，获得了下列结论：实际网络的度分布指数不会低于1；度分布指数介于1～2之间的复杂网络中存在数量较多的HUB节点，其边数与节点数之间的关系是非线性的，节点数的增加将导致边数的大幅度增加；度分布指数介于2～3之间的复杂网络中存在一定数量的HUB节点，其边数与节点数之间的关系是线性的，大多数受成本制约的网络属于这种类型；度分布指数大于3的复杂网络近似于均质网络；度分布指数3构成了复杂网络中病毒防治方式的临界点。

Purpose. It is already known that the topological structure and propagation dynamics of complex network are closely dependent on its degree distribution, which, in its turn, is completely determined by its degree distribution exponent. Most of real networks are found, by empirical study, to have the degree distribution exponents located between 2 and 3. But, to our best knowledge, the theoretical basis of this important empirical knowledge is as yet lacking. This paper aims to provide such a theoretical basis. In the full paper, we explain in detail our theoretical research; in this abstract we just mention that our theoretical conclusions are reached through derivation and discussion of eqs. （1） through （10） in the full paper. The following conclusions are obtained： （1） the degree distribution exponents of real networks cannot be less than 1; （2） there exist plenty of hub nodes for complex networks whose degree distribution exponents are between 1 and 2, the edges and nodes have nonlinear relations, and the increase of nodes will result in much more increase of edges; （3） for complex networks whose degree distribution exponents are between 2 and 3, the edges are linearly dependent on the nodes, and most networks, whose constructions are heavily controlled by cost, are of this kind; （4） complex networks whose degree distribution exponents are greater than 3 are homogeneous; （5） the degree distribution exponent 3 is a critical point for the prevention of virus propagation of complex network.