  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.
