## Explosive Percolation in Random NetworksThis thesis is devoted to the study of the Bohman-Frieze-Wormald percolation model, which exhibits a discontinuous transition at the critical threshold, while the phase transitions in random networks are originally considered to be robust continuous phase transitions. The underlying mechanism that leads to the discontinuous transition in this model is carefully analyzed and many interesting critical behaviors, including multiple giant components, multiple phase transitions, and unstable giant components are revealed. These findings should also be valuable with regard to applications in other disciplines such as physics, chemistry and biology. |

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### Contents

1 | |

2 Discontinuous Explosive Percolation with Multiple Giant Components | 9 |

3 Deriving an Underlying Mechanism for Discontinuous Percolation Transitions | 17 |

4 Continuous Phase Transitions in Supercritical Explosive Percolation | 28 |

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### Common terms and phrases

accepted edges Achlioptas process algorithm asymptotically behavior BFW model C1 and C2 Chen complete graph complex networks continuous percolation D’Souza denote the number direct growth Discontinuous percolation transitions discontinuous phase transition discontinuous transition Dorogovtsev edge density evolution explosive percolation finite fraction of accepted fraction of nodes function giant components emerge Goltsev growth by overtaking Havlin Herrmann isolated nodes jump in C1 k-core Kahng largest jump Lett link density macroscopic components merge multiple giant components Nagler Newman order parameter Percolation in Random percolation models percolation phase transitions Percolation Theory percolation threshold percolation tran percolation transition point Phys positive constant power law random graphs random networks sampled scale-free networks scaling second largest component single edge site percolation small-world networks square lattice stable giant components Strogatz strongly discontinuous studied supercritical regime thermodynamic limit Tricritical point underlying mechanism uniformly at random Ziff