Degree and component size distributions in generalized uniform recursive tree

Abstract

We propose a generalized model for uniform recursive tree (URT) by introducing an imperfect growth process, which may generate disconnected components (clusters). The model undergoes an interesting phase transition from a singly connected network to a graph consisting of fully isolated nodes. We investigate the distributions of degree and component sizes by both theoretical predictions and numerical simulations. For the nontrivial cases, we show that the network has an exponential degree distribution while its component size distribution follows a power law, both of which are related to the imperfect growth process. We also predict the growth dynamics of the individual components. All analytical solutions are successfully contrasted with computer simulations.

4 pages, 3 figures

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• Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks
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• Structural and spectral properties of a family of deterministic recursive trees: Rigorous solutions