环形holonomy诱导高阶约束并通过扭曲拉普拉斯谱控制远程同步转变
Cycle holonomy captures higher-order compatibility constraints in remote synchronization
AI总结：
研究通过扭曲拉普拉斯谱分析环形holonomy对远程同步转变的影响，揭示拓扑摩擦与网络上同调之间的联系。
摘要：
高阶相互作用网络通常用超图或单纯复形建模，其中相互作用涉及超过两个节点。本文证明在图的1-骨架上，当相互作用具有非平凡拓扑结构时，会自然产生有效的高阶动态约束。我们研究了边相滞后建模为U(1)-valued连接的相振子动力学，这种结构诱导出梯度Sakaguchi-Kuramoto型流和相关的扭曲拉普拉斯算子，其谱依赖于连接的上同调类。证明扭曲拉普拉斯算子存在零模当且仅当连接是上同调平凡的，即所有环形holonomy消失。因此，同步受制于内在拓扑摩擦而非局部配对不匹配。推导出扭曲拉普拉斯算子的最小特征值与holonomy的大小成比例，其谱转变准确预测了随着摩擦增加相锁定状态的稳定性丧失。对于常数相滞后特例，我们解析推导出临界转变点α_c=π/3对于五边形环，与之前报告的数值阈值定量一致。结果建立了动态摩擦与网络上同调的谱框架，并显示远程同步转变受环形拓扑约束影响。
英文摘要：
Higher-order interactions have typically been modeled using hypergraphs or simplicial complexes, where interactions explicitly involve more than two nodes. Here we demonstrate that effective higher-order dynamical constraints emerge naturally on ordinary graphs, provided the interaction carries nontrivial topological structure. We study a gauge-coupled phase model with edge phase lags whose accumulation around closed loops produces gauge-invariant mismatches. We show that the associated twisted Laplacian admits a zero mode if and only if all cycle holonomies vanish. Consequently, global compatibility is obstructed not by local pairwise mismatches, but by intrinsic topological frustration on cycles. We then connect this framework to the symmetric Sakaguchi--Kuramoto model, whose local coupling law differs from the gauge-coupled model but whose node phases impose cycle closure on physical phase differences. For cactus graphs, path mismatches induced by the symmetric lag can be represented through associated cycle holonomies, providing a static spectral encoding of their global residual incompatibility. Our results establish a spectral framework linking frustration to cycle-level constraints and identify cycle holonomy as a local-to-global diagnostic of path incompatibility in synchronization dynamics.
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