BA指数网络结构特性精确计算

Exact computations for structural characteristics of BA exponential networks

度分布和平均路径长度是复杂网络的两个重要结构特性.对于随机增长的网络,其平均路径长度一般主要通过计算机模拟给出数值结果,尚无一个普适的解析计算方法.为此首先利用主方程的方法对BA随机指数网络的度分布进行解析推导,与B arabás i等的结果相比所得度分布与计算机模拟值更接近.然后对BA随机指数网络和BA确定性指数网络的平均路径长度进行解析计算,所得结果与经典的ER随机图相似,即平均路径长度以网络大小的对数形式增长.此外,对BA随机指数网络平均路径长度的模拟值与解析计算结果相吻合.最后,对BA确定性指数网络的度分布与直径进行了解析计算,并对两个网络的结构特性作了比较.比较结果表明,BA随机指数网络和BA确定性指数网络的结构性质虽然存在量上的差异,但从定性角度来说,其拓扑结构是相同的.

Degree distribution and average path length are two important structural characteristics of complex networks. For randomly growing networks, there has not been a generic analytical computation technique for their average path length, which is obtained mainly by computer simulations. To solve this problem, firstly, the expression of degree distribution for the random exponential BA network is derived analytically by master-equation approach, which is more consistent with the simulations than that given by Barabási, et al. Then the emphasis switches to the computation of the average path length for both random and deterministic exponential BA networks. The obtained average path length is similar to the counterpart of ER random graph, which increases logarithmically with the network size. Also, simulations are done for the properties of the random situation, which are in good agreement with the analytical results. Furthermore, the degree distribution and diameter of the deterministic exponential BA network are found exactly. Finally, comparative research is done on the structural characteristics of the two networks. The obtained results show that the two networks have quantitatively different topological properties, while they are qualitatively similar.