Random networks with preferential growth and vertex death

A dynamic model for a random network evolving in continuous time is defined where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function $b$ of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function $d$ of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution $\{p_k\}$ is derived and analyzed for a number of specific choices of $b$ and $d$. When $b(i)=i+\alpha$ and $d(i)=\beta$ -- that is, linear preferential attachment for the newborn and random deaths -- then $p_k\sim k^{-(2+\alpha)}$. When $b(i)=i+1$ and $d(i)=\beta(i+1)$, with $\beta<1$, then $p_k\sim (1+\beta)^{-k}$, that is, if also the death rate is proportional to the fitness, then the power law distribution is lost. Furthermore, when $b(i)=i+1$ and $d(i)=\beta(i+1)^\gamma$, with $\beta,\gamma<1$, then $\log p_k\sim -k^\gamma$ -- a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.