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The article deals with two classes of growing random graphs following thepreferential attachment rule with a linear weight function, L-graphs, andhybrid Pennock graphs. We determine the exact final vertex degree distributionand the exact final two-dimensional arcs degree distributions of graphs underconsideration. The study proves that each hybrid Pennock graph is isomorphic toa certain L graph and that the converse does not hold since there are noPennock graphs isomorphic to L graphs with negative displacements in theexpression for the linear weight function. A formula is derived that makes itpossible to determine the weight functions, which are used to generate graphswith the required asymptotic power-law vertex degree distribution. Thereliability of the obtained results is confirmed by calculations using accuratenumerical methods and simulation modeling, i.e. by direct generating of thegraphs. The practical value of the results is demonstrated by an example oftheir effective application for accurate calibration of a growing graph thatsimulates a network of Internet at the level of autonomous systems.

Vertex degree distribution and arc endpoints degree distribution of graphs with a linear rule of preferential attachment and Pennock graphs