Cluster structure of optimal solutions in bipartitioning of small worlds

TL;DR

This paper investigates how the structure of optimal bipartitions in small-world networks depends on the underlying lattice, revealing finite clusters in 1D and stripe-like patterns in 2D, with implications for symmetry breaking.

Contribution

It demonstrates that bipartitioning in small worlds varies with the lattice, showing finite clusters in 1D and stripe regimes in 2D, contrasting mean-field expectations.

Findings

Optimal partitions in 1D are finite clusters for any added links.

In 2D, small added links produce stripe-like partitions, which become disordered with more links.

Replica symmetry is broken in 1D but preserved with degeneracy in 2D.

Abstract

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.