  Higher-order networks are able to capture the many-body interactions present
in complex systems and to unveil new 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, overcoming thus 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 we 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.
