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.
A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M -clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M -clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.
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