Wang Runyue, Muolo Riccardo, Carletti Timoteo, Bianconi Ginestra
School of Mathematical Sciences, <a href="https://ror.org/026zzn846">Queen Mary University of London</a>, London E1 4NS, United Kingdom.
Department of Systems and Control Engineering, <a href="https://ror.org/0112mx960">Tokyo Institute of Technology</a>, 2 Chome-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan.
Phys Rev E. 2024 Jul;110(1-1):014307. doi: 10.1103/PhysRevE.110.014307.
Higher-order networks are able to capture the many-body interactions present in complex systems and to unveil fundamental phenomena revealing the rich interplay between topology, geometry, and dynamics. Simplicial complexes are higher-order networks that encode higher-order topology and dynamics of complex systems. Specifically, simplicial complexes can sustain topological signals, i.e., dynamical variables not only defined on nodes of the network but also on their edges, triangles, and so on. Topological signals can undergo collective phenomena such as synchronization, however, only some higher-order network topologies can sustain global synchronization of topological signals. Here we consider global topological synchronization of topological signals on weighted simplicial complexes. We demonstrate that topological signals can globally synchronize on weighted simplicial complexes, even if they are odd-dimensional, e.g., edge signals, thus overcoming a limitation of the unweighted case. These results thus demonstrate that weighted simplicial complexes are more advantageous for observing these collective phenomena than their unweighted counterpart. In particular, we present two weighted simplicial complexes: the weighted triangulated torus and the weighted waffle. We completely characterize their higher-order spectral properties and demonstrate that, under suitable conditions on their weights, they can sustain global synchronization of edge signals. Our results are interpreted geometrically by showing, among the other results, that in some cases edge weights can be associated with the lengths of the sides of curved simplices.
高阶网络能够捕捉复杂系统中存在的多体相互作用，并揭示拓扑、几何和动力学之间丰富相互作用的基本现象。单纯复形是对复杂系统的高阶拓扑和动力学进行编码的高阶网络。具体而言，单纯复形能够维持拓扑信号，即不仅定义在网络节点上，还定义在其边、三角形等上的动力学变量。拓扑信号可以经历诸如同步之类的集体现象，然而，只有一些高阶网络拓扑能够维持拓扑信号的全局同步。在这里，我们考虑加权单纯复形上拓扑信号的全局拓扑同步。我们证明，即使拓扑信号是奇数维的，例如边信号，它们也能够在加权单纯复形上实现全局同步，从而克服了无加权情况下的一个限制。因此，这些结果表明，加权单纯复形在观察这些集体现象方面比其无加权对应物更具优势。特别是，我们提出了两种加权单纯复形：加权三角环面和加权华夫饼。我们完全刻画了它们的高阶谱性质，并证明在其权重的适当条件下，它们能够维持边信号的全局同步。我们通过展示，在其他结果中，在某些情况下边权重可以与弯曲单纯形的边长相关联，从而对我们的结果进行几何解释。
