We consider a new model of the web graph and related networks. The model is motivated by the copying models of the web graph, where new nodes copy the link structure of existing nodes, and a certain number of additional random links are introduced. Our model parametrizes the number of random links, thereby allowing for the analysis of threshold behaviour. We consider infinite limits of graphs generated by our model, and compare properties of these limits with orientations of the infinite random graph. We analyze the power law behaviour of the in-degree distribution of graphs generated by our model.
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