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.

中文翻译：

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小世界二分时最优解的簇结构

使用模拟退火，我们检查通过将一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二分。在小世界上定义的模型通常表现出平均场行为，而不管底层晶格如何。我们的工作表明，小世界的二分确实取决于底层的格子。模拟表明，对于一维小世界，最佳分区是附加链接的任何部分的有限大小集群。在二维情况下，我们观察到两种情况：当附加链接的比例足够小时，最佳分区具有条状形状，随着最佳分区变得无序，大量附加链接会丢失。一些论据，将附加链接解释为热激发并参考 Ising 模型的热力学，提出了对这种行为的定性解释。重叠的直方图表明一维小世界中的副本对称性被破坏。在二维情况下，副本对称性似乎成立，但条纹状分区有一些额外的简并性。