Robustness of the in-degree exponent for the world-wide web

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 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 the in-degree 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 variance $\sigma^2$. Using both analytic treatment and numerical simulations, we show that while the out-degree exponent $\gamma_{\rm out}$ depends on the parameters, the in-degree exponent $\gamma_{\rm in}$ has two distinct values, $\gamma_{\rm in}=2$ 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 $\gamma_{\rm in}=2$ for the www.