Recently several authors have proposed stochastic evolutionary models for thegrowth of complex networks that give rise to power-law distributions. Thesemodels are based on the notion of preferential attachment leading to the ``richget richer'' phenomenon. Despite the generality of the proposed stochasticmodels, there are still some unexplained phenomena, which may arise due to thelimited size of networks such as protein and e-mail networks. Such networks mayin fact exhibit an exponential cutoff in the power-law scaling, although thiscutoff may only be observable in the tail of the distribution for extremelylarge networks. We propose a modification of the basic stochastic evolutionarymodel, so that after a node is chosen preferentially, say according to thenumber of its inlinks, there is a small probability that this node will bediscarded. We show that as a result of this modification, by viewing thestochastic process in terms of an urn transfer model, we obtain a power-lawdistribution with an exponential cutoff. Unlike many other models, the currentmodel can capture instances where the exponent of the distribution is less thanor equal to two. As a proof of concept, we demonstrate the consistency of ourmodel by analysing a yeast protein interaction network, the distribution ofwhich is known to follow a power law with an exponential cutoff.
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