Topological Reorganization and Coordination-Controlled Crossover in Synchronization Onset on Regular Lattices

Abstract

The transition to global synchronization in coupled dynamical systems is governed by the interplay between coupling strength and structural topology. Although abrupt, first-order-like synchronization transitions have been extensively reported in heterogeneous networks, it is unclear whether comparable accelerated onset behavior can emerge purely from coordination geometry in spatially homogeneous, regular lattices. In this study, we investigate large-scale ($N=10^5$) stochastic Stuart-Landau oscillator networks defined on regular lattices with controlled coordination number. Using topological data analysis (TDA), simplicial-complex characterization, and optimal-transport-based geometric diagnostics, we identify a coordination-controlled crossover in synchronization onset dynamics at approximately $z_{c} \approx 7$ within the class of regular lattices considered. Low-coordination lattices ($z < z_{c}$) exhibit persistent $H_2$ topological features in the dynamical amplitude field that correlate with delayed coherence and surface-limited propagation. In contrast, higher-coordination lattices ($z > z_{c}$) display rapid fragmentation of these features, reduced interface roughness, and predominantly positive Ricci curvature. This is consistent with enhanced path redundancy and improved transport efficiency. In this regime, the global order parameter exhibits accelerated exponential-like growth during the onset stage. Throughout this work, abrupt synchronization refers specifically to this exponential onset behavior rather than to thermodynamic first-order hysteresis. Our results demonstrate that increasing coordination density induces a qualitative reorganization of higher-order topological structure that strongly correlates with synchronization efficiency in regular lattice systems.

Figures (7)

• Figure 1: Geometric expansion vs. abrupt synchronization. (a) Log-log plot showing early-stage growth. Low-$z$ lattices follow geometric power laws ($t^2, t^3$). (b) Semi-log plot. fcc ($z=12$) exhibits exponential ascent ($e^{\alpha t}$), while sc ($z=6$) shows prolonged inertial delay.
• Figure 2: Topological persistence barcodes. Comparison of homological features ($H_1, H_2$). The 3D sc lattice (c) exhibits persistent trapping voids, whereas the 3D fcc lattice (e) shows transient shattering.
• Figure 3: Temporal evolution of void fragment count. fcc shows rapid shattering (peak), while sc shows surface erosion (flat).
• Figure 4: Geometric origin of speed. (a) Wavefront roughness $W(t)$. Here, sc is rough (high friction), fcc is smooth. (b) Simplicial density. High density in fcc correlates with low roughness.
• Figure 5: Structural crossover in geometric synchronizability $\chi$ vs. coordination number $z$. A clear transition occurs at $z_c \approx 7$, separating the trapping phase ($z<7$, green) from the shattering phase ($z>7$, red). The dashed curve represents a phenomenological sigmoid fit $\chi(z) = \chi_{0} + A / [1 + \exp(-k(z - z_c))]$ with parameters $\chi_{0} \approx 0.05$, $A \approx 0.95$, $k \approx 1.5$, and $z_c \approx 7$. Note that only five crystallographically distinct regular lattice structures exist in 2D and 3D (square, triangular, sc, bcc, fcc), limiting quantitative precision of the crossover location. Physically, this threshold is reminiscent of the rigidity percolation transition in 3D networks Jacobs1995, where a minimal coordination is required to propagate global mechanical stability.