无标度网络度秩指数的变化范围

实证研究表明,绝大多数复杂网络的结点的度分布服从幂律分布,该幂律分布的幂指数的绝对值(度分布指数)介于2和3之间．然而,至今尚未发现为什么度分布指数介于2和3之间的研究结果．本文证明了度分布指数大于2,从而部分回答了上述问题．为此,本文引进度秩指数,并给出了度秩指数和度分布指数之间的关系．通过对度秩指数与网络结构熵之间的关系的刻画,发现了度秩指数与网络结构熵以及网络规模之间的函数依赖关系,从而最终证明了度秩指数的临界值趋于1,并给出了仿真结果．

Range of degree-rank exponent of scale-free networks

Empirical study shows that most of complex networks are scale-free, and the node degrees of which obey the power-law distribution with the absolute value of the corresponding power exponent (known as degree distribution exponent) lying between 2 and 3. However, there is no research result on why the degree distribution exponent lies between 2 and 3 so far. The authors prove that the degree distribution exponent is greater than 2 and thus partially solve the problem mentioned above. First, the degree-rank exponent is introduced, the relation between the degree-rank exponent and the degree distribution exponent is also given. Through the characterization on the relation between the degree-rank exponent and the network structural entropy, functional dependent relations among the degree-rank exponent, the network structural entropy and the number of nodes of the network are then obtained and the result that the degree-rank exponent approaches one is proved. Finally, simulation results are given to illustrate the theoretical result.