  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.
