The degree sequence of a scale-free random graph process
Random Structures & Algorithms
Preferential attachment in online networks: measurement and explanations
Proceedings of the 5th Annual ACM Web Science Conference
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We consider a tree that grows randomly in time. Each time a newvertex appears, it chooses exactly one of the existing vertices andattaches to it. The probability that the new vertex chooses vertexx is proportional to w(deg(x)), a weightfunction of the actual degree of x. The weight functionw : ℕ ➝ ℝ+ is the parameter ofthe model.In [4] and [11] the authors derive the asymptoticdegree distribution for a model that is equivalent to the specialcase, when the weight function is linear. The proof thereinstrongly relies on the linear choice of w.Using well-established results from the theory of generalbranching processes we give the asymptotical degree distributionfor a wide range of weight functions. Moreover, we provide theasymptotic distribution of the tree itself as seen from a randomlyselected vertex. The latter approach gives greater insight to thelimiting structure of the tree.Our proof is robust and we believe that the method may be usedto answer several other questions related to the model. It relieson the fact that considering the evolution of the random tree incontinuous time, the process may be viewed as a general branchingprocess, this way classical results can be applied. © 2006Wiley Periodicals, Inc. Random Struct. Alg., 2007