Luo Ziting, Chen Wei, Nagler Jan
LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China.
Institute of Artificial Intelligence, Beihang University, Beijing 100191, China.
Phys Rev E. 2023 Sep;108(3-1):034108. doi: 10.1103/PhysRevE.108.034108.
We study explosive percolation processes on random graphs for the so-called product rule (PR) and sum rule (SR), in which M candidate edges are randomly selected from all possible ones at each time step, and the edge with the smallest product or sum of the sizes of the two components that would be joined by the edge is added to the graph, while all other M-1 candidate edges are being discarded. These two rules are prototypical "explosive" percolation rules, which exhibit an extremely abrupt yet continuous phase transition in the thermodynamic limit. Recently, it has been demonstrated that PR and SR belong to the same universality class for two competing edges, i.e., M=2. Here we investigate whether the claimed PR-SR universality is valid for higher-order models with M larger than 2. Based on traditional finite-size scaling theory and largest-gap scaling, we obtain the percolation threshold and the critical exponents of the order parameter, susceptibility, and the derivative of entropy for PR and SR for M from 2 to 9. Our results strongly suggest PR-SR universality, for any fixed M.
我们研究了随机图上所谓的乘积规则（PR）和求和规则（SR）的爆发性渗流过程，其中在每个时间步从所有可能的边中随机选择M条候选边，并且将边所连接的两个组件大小的乘积或和最小的边添加到图中，而所有其他M - 1条候选边则被丢弃。这两条规则是典型的“爆发性”渗流规则，在热力学极限下表现出极其突然但连续的相变。最近，已经证明对于两个竞争边（即M = 2），PR和SR属于同一普适类。在此我们研究对于M大于2的高阶模型，所宣称的PR - SR普适性是否成立。基于传统的有限尺寸标度理论和最大间隙标度，我们得到了M从2到9时PR和SR的渗流阈值以及序参量、磁化率和熵的导数的临界指数。我们的结果强烈表明对于任何固定的M，PR - SR普适性成立。
