Zhang Yang, Wei Wei, Guo Binghui, Zhang Renquan, Zheng Zhiming
LMIB and School of Mathematics and Systems Sciences, Beihang University, 100191 Beijing, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 1):051103. doi: 10.1103/PhysRevE.86.051103. Epub 2012 Nov 5.
Percolation is one of the most widely studied models in which a unique giant cluster emerges after the phase transition. Recently, a new phenomenon, where multiple giant clusters are observed in the so called Bohman-Frieze-Wormald (BFW) model, has attracted much attention, and how multiple giant clusters could emerge in generic percolation processes on random networks will be discussed in this paper. By introducing the merging probability and inspecting the distinct mechanisms which contribute to the growth of largest clusters, a sufficient condition to generate multiple stable giant clusters is given. Based on the above results, the BFW model and a multi-Erdös-Rényi (ER) model given by us are analyzed, and the mechanism of multiple giant clusters of these two models is revealed. Furthermore, large fluctuations are observed in the size of multiple giant clusters in many models, but the sum size of all giant clusters exhibits self-averaging as that in the size of unique giant cluster in ordinary percolation. Besides, the growth modes of different giant clusters are discussed, and we find that the large fluctuations observed are mainly due to the stochastic behavior of the evolution in the critical window. For all the discussion above, numerical simulations on the BFW model and the multi-ER model are done, which strongly support our analysis. The investigation of merging probability and the growth mechanisms of largest clusters provides insight for the essence of multiple giant clusters in the percolation processes and can be instructive for modeling or analyzing real-world networks consisting of many large clusters.
渗流是研究最为广泛的模型之一，在相变之后会出现一个独特的巨型簇。最近，一种新现象引起了广泛关注，即在所谓的博曼 - 弗里兹 - 沃尔默德（BFW）模型中观察到多个巨型簇，本文将讨论在随机网络上的一般渗流过程中如何出现多个巨型簇。通过引入合并概率并考察促成最大簇增长的不同机制，给出了产生多个稳定巨型簇的充分条件。基于上述结果，对我们给出的BFW模型和多厄尔多斯 - 雷尼（ER）模型进行了分析，揭示了这两个模型中多个巨型簇的形成机制。此外，在许多模型中，多个巨型簇的大小存在大幅波动，但所有巨型簇的总大小呈现出自平均特性，这与普通渗流中单个巨型簇大小的情况类似。此外，还讨论了不同巨型簇的增长模式，我们发现观察到的大幅波动主要归因于临界窗口内演化的随机行为。针对上述所有讨论，对BFW模型和多ER模型进行了数值模拟，有力地支持了我们的分析。对合并概率和最大簇增长机制的研究为渗流过程中多个巨型簇的本质提供了见解，并且对由许多大簇组成的现实世界网络的建模或分析具有指导意义。
