The Degree Profile in Some Classes of Random Graphs that Generalize Recursive Trees

We study the degree profile for a number of classes of random graphs that arise as generalizations of recursive trees, including random circuits and random recursive trees endowed with the power of choice. We investigate the distribution of the degrees of nodes that appear in various stages of the insertion process in each of these graph types. For these classes, we will see phase transitions in degrees depending on the stage—early stages are associated with normal distributions, intermediate stages are associated with the Poisson distribution and in the late stages the degrees become degenerate.