论文标题
最佳解决方案的群集结构
Cluster structure of optimal solutions in bipartitioning of small worlds
论文作者
论文摘要
使用模拟退火，我们检查了通过将一小部分随机选择的链接添加到一维链或方形晶格中获得的小世界的两部分。在小世界上定义的模型通常表现出平均场行为，无论其基础晶格如何。我们的作品表明，小世界的两部分确实取决于基础晶格。模拟表明，对于一维小世界，最佳分区是有限尺寸的簇，适用于其他链接的任何部分。在二维情况下，我们观察到了两个机制：当其他链接的分数足够小时，最佳分区具有类似条纹的形状，随着最佳分区变得无序，则丢失了更多的其他链接。一些论点将其他链接解释为热激发，并指伊辛模型的热力学，提出了对这种行为的质量解释。重叠的直方图表明，在一维小世界中，复制对称性被打破。在二维情况下，复制对称性似乎可以保持，但具有类似条纹的分区的额外变性。
Using a simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behaviour, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitatitve explanation of such a behaviour. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold but with some additional degeneracy of stripe-like partitions.
