Kahng B, Park Y, Jeong H
School of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Oct;66(4 Pt 2):046107. doi: 10.1103/PhysRevE.66.046107. Epub 2002 Oct 10.
We consider a stochastic model for directed scale-free networks following power laws in the degree distributions in both incoming and outgoing directions. In our model, the number of vertices grow geometrically with time with a growth rate p. At each time step, (i) each newly introduced vertex is connected to a constant number of already existing vertices with the probability linearly proportional to in-degree distribution of a selected vertex, and (ii) each existing vertex updates its outgoing edges through a stochastic multiplicative process with mean growth rate of outgoing edges g and its variance sigma(2). Using both analytic treatment and numerical simulations, we show that while the out-degree exponent gamma(out) depends on the parameters, the in-degree exponent gamma(in) has two distinct values, gamma(in)=2 for p>g and 1 for p<g, independent of different parameters values. The latter case has logarithmic correction to the power law. Since the vertex growth rate p is larger than the degree growth rate g for the World-Wide Web (WWW) nowadays, the in-degree exponent appears robust as gamma(in)=2 for the WWW.
我们考虑一个有向无标度网络的随机模型，该模型在入度和出度分布上均遵循幂律。在我们的模型中，顶点数量随时间呈几何增长，增长率为p。在每个时间步，（i）每个新引入的顶点以与所选顶点入度分布成线性比例的概率连接到固定数量的已存在顶点，并且（ii）每个现有顶点通过一个随机乘法过程更新其出边，该过程的出边平均增长率为g，方差为sigma(2)。通过解析处理和数值模拟，我们表明，虽然出度指数gamma(out)取决于参数，但入度指数gamma(in)有两个不同的值，当p>g时，gamma(in)=2；当p<g时，gamma(in)=1，与不同的参数值无关。后一种情况对幂律有对数修正。由于如今万维网（WWW）的顶点增长率p大于度增长率g，所以对于WWW来说，入度指数似乎稳定为gamma(in)=2。
