Abstract
Using 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 behavior, 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 a 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 qualitative explanation of such a behavior. 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.
Highlights
Optimization problems draw considerable interest from computer scientists, engineers, economists, and mathematicians
Our results show that bipartitioning of small worlds can be to some extent understood by referring to the thermodynamic behavior of the Ising model on the underlying regular network
We present the results of our calculations obtained for small worlds on a linear chain (d = 1) and a square lattice (d = 2)
Summary
Optimization problems draw considerable interest from computer scientists, engineers, economists, and mathematicians. Statistical mechanics approaches exploit the analogy with the Ising model and are fruitful in the random graph version of this problem Such a version was studied numerically using simulated annealing [6,7] or an extremal optimization [8], but important analytical results were obtained using the replica method [9,10,11], a technique that was primarily developed for studying disordered systems. Let us notice that bipartitioning of regular Cartesian lattices is nearly trivial and results in optimal partitions being simple and compact clusters such as sections (d = 1) or stripes (d = 2) These simple partitions are ground-state configurations of the Ising model subject to the constraint of zero total magnetization.
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