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  • 1 Department of Probability Theory and Statistics, Faculty of Science, Eötvös Loránd University, Pázmány P. sétány. 1/C, H-1117 Budapest, Hungary
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In several scale free graph models the asymptotic degree distribution and the characteristic exponent change when only a smaller set of vertices is considered. After recalling the sufficient conditions for the existence of asymptotic local degree distribution [1], several random graph models are presented that satisfy these assumptions. We show the necessity of the main conditions by constructing counterexamples.