A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of N vertices (N ≥ 3). A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of the interactions of the vertex. The asymptotic behaviour of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2, ∞) can be achieved. The proofs are based on discrete time martingale theory.
A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.
Abstract. In this paper two modifications of Kolchin's generalized allocation scheme are studied. Results known for Kolchin's scheme are extended to the new models. Representation theorems, strong laws of large numbers and local limit theorems are obtained. In the proofs some general inequalities are used.
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