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One of the most studied random graphs is G(n, p), which has n vertices that can be taken as the integers 1,..., n, and where each pair of vertices is connected by an edge with probability p, independently of all other edges. We consider the case p = c/n for some constant c> 0, and let n → ∞. The degree of a given vertex has a binomial distribution Bi(n−1, c/n) ≈ Po(c). This is a strongly concentrated distribution with an exponentially decreasing tail. Many graphs from “real life” has much larger tails, for example power-law tails, and it is therefore important to study also random graph models with such behaviour. 1 We will describe one class of random graphs that generalize G(n, c/n) but also allow many less homogeneous examples, for example natural examples of ‘scale-free ’ random graphs, where the degree distribution has a powerlaw tail. We believe that when it comes to modelling real-world graphs with, for example, observed power-laws for vertex degrees, our model provides an interesting and flexible alternative to existing models. Nevertheless, we will see that many properties of G(n, c/n) extend to these random graphs. In particular, we consider the question whether there exist a giant component or not, and we will, typically, find a phase transition similar to what happens for G(n, c/n). There are, however, some interesting twists for some examples. We are interested in graphs with a large number of vertices, and in particular in asymptotics as the number tends to infinity. The graphs we consider are such that the average degree stays bounded, so they are rather sparse

Year: 2006

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