Exclusive oscillator interactions generate globally coherent phase arrangements: insights from benchmark traveling salesman problems

Various oscillatory phenomena are observed in nature. Oscillatory interactions can contribute to the self-organization of globally coherent relationships and may play important roles in information processing, control, and communication in biological systems. However, the mechanisms by which such relationships are generated remain unclear. This study used the traveling salesman problem, a classic combinatorial optimization problem, as a benchmark to investigate whether exclusive interactions between Kuramoto phase oscillators can generate globally coherent phase arrangements through self-organization. We assigned cities to individual oscillators and introduced mutual repulsive interactions based on inter-city distances and phase differences. Our analyses showed that the oscillators frequently converged to phase orders corresponding to the shortest tour, although two-cluster solutions were also observed for some distance sets. Even when two-cluster solutions emerged, adding a new oscillator at a sufficiently large distance from the existing oscillators markedly increased the probability of obtaining the shortest-tour solution. In a special case, convergence to the shortest tour was also successfully demonstrated in simulations with up to 50 oscillators. These findings suggest that exclusive oscillator interactions can generate globally coherent phase arrangements through self-organization and provide a simple framework for investigating biologically inspired collective dynamics. Although a precise mathematical understanding of these dynamics remains a subject for future investigation, the present framework provides a platform for exploring oscillator-based computation through self-organized phase dynamics.