  We propose a scale-free network model with a tunable power-law exponent. The
Poisson growth model, as we call it, is an offshoot of the celebrated model of
Barab\'{a}si and Albert where a network is generated iteratively from a small
seed network; at each step a node is added together with a number of incident
edges preferentially attached to nodes already in the network. A key feature of
our model is that the number of edges added at each step is a random variable
with Poisson distribution, and, unlike the Barab\'{a}si-Albert model where this
quantity is fixed, it can generate any network. Our model is motivated by an
application in Bayesian inference implemented as Markov chain Monte Carlo to
estimate a network; for this purpose, we also give a formula for the
probability of a network under our model.
