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 ${\gamma }_{\mathrm{out}}$ depends on the parameters, the in-degree exponent ${\gamma }_{\mathrm{in}}$ has two distinct values, ${\gamma }_{\mathrm{in}}=2\mathrm{}$ 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 ${\gamma }_{\mathrm{in}}=2\mathrm{}$ for the WWW.

• Received 14 December 2001

DOI:https://doi.org/10.1103/PhysRevE.66.046107