Bose-Einstein condensation in random directed networks

We consider the phenomenon of Bose-Einstein condensation in a random growing directed network. The network grows by the addition of vertices and edges. At each time step the network gains a vertex with probabilty $p$ and an edge with probability $1-p$. The new vertex has a fitness $(a,b)$ with probability $f(a,b)$. A vertex with fitness $(a,b)$, in-degree $i$ and out-degree $j$ gains a new incoming edge with rate $a(i+1)$ and an outgoing edge with rate $b(j+1)$. The Bose-Einstein condensation occurs as a function of fitness distribution $f(a,b)$.