Local pair correlations and network geometry govern collective oscillations in nonreciprocal Ising networks

Nonreciprocal many-body systems can sustain spontaneous, time-crystal-like global oscillations, making their quantitative prediction an important problem in nonequilibrium physics. Here we propose nearest-neighbor spin-coupling theory (NSC) for the two-spin nonreciprocal Ising model. By retaining pair correlations between adjacent nodes, NSC predicts global oscillations through nine independent local probabilities for their joint spin states. We compare NSC with mean-field theory (MF) and independent-node theory (ID) using extensive Monte Carlo simulations across system sizes and coupling strengths. NSC recovers oscillations missed by MF at the reference coupling and corrects ID’s overestimation of their amplitude and angular momentum. On edge-swapped networks, it predicts amplitude, angular momentum, and period with a 2.1% aggregate mean absolute relative error across the common oscillatory regime. We then compare periodic square lattices and degree-preserving edge-swapped networks with identical local interaction parameters and coordination number. Oscillation amplitude and angular momentum decay strongly with increasing system size on the square lattice but remain nearly size independent on edge-swapped networks. Spatial phase maps reveal phase cancellation between lattice domains and nearly uniform phase alignment on edge-swapped networks. Our findings provide a quantitative basis for understanding how local pair correlations and network geometry govern collective oscillations in nonreciprocal many-body systems.