  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.
