具有幂率度分布的因特网平均最短路径长度估计

针对具有幂律度分布的复杂网络的平均最短路径长度进行了研究.给出了一个计算因特网平 均最短路径长度的公式.提出因特网的整体构造实质是以最小代价换取最大收益，从此出 发通过对因特网这类复杂网络平均最短路径长度影响因素分析，推断出网络最短路径长度分 布P(l)的基本性质，进而构造了一个只含参数α的P(l)的简洁形式，直接打通了拓扑具有幂 律度分布的因特网的度分布P(k)与网络最短路径长度分布P(l)之间的关系.然后说明了导出 的公式的物理意义，即参数α代表网络节点连接方式和网络的总边数对平均最短路径长 度的影响.此公式意味着只要掌握幂律P(k)～k-α中α值，就可以直接计算相应网 络的平均最短路径长度.通过对已知数据的计算，验证了公式的有效性.阐明了参数 α对网络设计的重要性和对网络功能的影响. 关键词：复杂网络幂律度分布平均最短路径长度

Average path length of Internet with power law degree distribution

The average path length of Internet with power law degree distribution is stud ied in this paper. The main work is to give a formula to compute the average pat h length . The essential of Internet or many other complex networks is to realize the least cost (the smallest total number of edges) and the best benefi t (the short average path length), which can be expressed as two optimization fa ctors to minimize the average degree and the average path length . By analyzing the main factor affecting , we find the main property of p ath length distribution P(l), and then give a formula of P(l) only inclu ding one parameter α. This work shows that P(l) and a direct relation with P(k),and α represent the effective link between nodes and the tota l number of networks on . This formula makes it possible that if we get the α of P(k)～k-α, we can calculate easily. The simulati on shows the formula is quite exact. Our method is successful. In the end, we di scuss that the parameter αis very important for Internet design.