Degree and component size distributions in generalized uniform recursive tree

TL;DR

This paper introduces a generalized uniform recursive tree model with an imperfect growth process, revealing phase transitions and analyzing degree and component size distributions through theory and simulations.

Contribution

It presents a novel generalized model for URT incorporating disconnection, and provides analytical and numerical analysis of degree and component size distributions.

Findings

Network exhibits exponential degree distribution.

Component sizes follow a power-law distribution.

Phase transition from connected to disconnected network.

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.