  Recently several authors have proposed stochastic evolutionary models for the
growth of complex networks that give rise to power-law distributions. These
models are based on the notion of preferential attachment leading to the ``rich
get richer'' phenomenon. Despite the generality of the proposed stochastic
models, there are still some unexplained phenomena, which may arise due to the
limited size of networks such as protein, e-mail, actor and collaboration
networks. Such networks may in fact exhibit an exponential cutoff in the
power-law scaling, although this cutoff may only be observable in the tail of
the distribution for extremely large networks. We propose a modification of the
basic stochastic evolutionary model, so that after a node is chosen
preferentially, say according to the number of its inlinks, there is a small
probability that this node will become inactive. We show that as a result of
this modification, by viewing the stochastic process in terms of an urn
transfer model, we obtain a power-law distribution with an exponential cutoff.
Unlike many other models, the current model can capture instances where the
exponent of the distribution is less than or equal to two. As a proof of
concept, we demonstrate the consistency of our model empirically by analysing
the Mathematical Research collaboration network, the distribution of which is
known to follow a power law with an exponential cutoff.
