Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit.

单纯复合物捕获了从大脑到社交网络的复杂系统的底层网络拓扑和几何结构。在这里，我们表明代数拓扑是捕获单纯复形的高阶动力学的基本工具。特别地，我们考虑拓扑信号，即定义在不同维度的单纯形上的动态信号，这里为了简单起见被视为节点和链接。我们表明节点和链接上定义的信号之间的耦合导致爆炸性拓扑同步，其中节点上定义的相位同时与不连续相变处的链接上定义的相位同步。我们研究真实连接组和单纯复合体和网络模型的模型。最后，