Lau Hon Wai, Paczuski Maya, Grassberger Peter
Complexity Science Group, University of Calgary, Calgary T2N 1N4, Canada.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jul;86(1 Pt 1):011118. doi: 10.1103/PhysRevE.86.011118. Epub 2012 Jul 17.
Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, adding links chosen randomly one by one from a set of predefined virtual links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a target cluster and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdös-Rényi networks), but most surprising were the results for two-dimensional (2D) lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3D) lattices--both of which are bipartite--are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z(2) symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k-partite graphs.
普通键渗流（OP）可被视为这样一个过程：团簇通过两两相连而生长，从一组预定义的虚拟链接中逐个随机选择链接来添加。相比之下，在凝聚渗流（AP）中，团簇通过随机选择一个目标团簇并将其与所有邻居相连而生长，邻居由同一组虚拟链接定义。先前的研究表明，对于几种类型的（虚拟）网络（线性链、树、厄多斯 - 雷尼网络），AP与OP处于不同的普适类，但最令人惊讶的是二维（2D）晶格的结果：虽然发现三角形晶格上的AP处于OP普适类，但它在正方形晶格上的行为却完全不同。在本文中，我们通过引入二分性来解释这种对普适性的显著违背。正方形晶格是二分图，而三角形晶格不是。与此一致，我们表明蜂窝晶格和简单立方（3D）晶格上的AP——这两者都是二分的——也不在OP普适类中。更确切地说，我们声称这种对普适性的违背基本上是由于在渗流阈值处自发破缺的Z(2)对称性。我们还讨论了二分随机网络上的AP以及k分图上AP的适当推广。
