2020/11/19 by Adam Lipowski, Antonio L. Ferreira, António L. Ferreira +1
Computer Science · Mathematics · Physics and Astronomy · #Cluster (spacecraft) #Degeneracy (biology) #Fraction (chemistry) #Histogram #Ising model #Opinion Dynamics and Social Influence #Regular polygon #Replica #Stochastic processes and statistical mechanics #Symmetry (geometry) #Theoretical and Computational Physics #cond-mat.stat-mech #cs.CC
paper · pdf · doi:10.3390/e22111319
published as Entropy 2020, 22, 1319 · 12 pages
arxiv created 2020/11/19 · openalex publication_date 2020/11/19 · arxiv updated 2020/11/20 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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.