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.
Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry can…
This Letter investigates the transition to synchronization of oscillator ensembles encoded by simplicial complexes in which pairwise and higher-order coupling weights alter with time through a rate-based adaptive mechanism inspired by the Hebbian learning rule. These simultaneously evolving disparate adaptive coupling weights lead to a phenomenon in which the in-phase synchronization is completely obliterated; instead, the anti-phase synchronization is originated. In addition, the onsets of antiphase synchronization and desynchronization are manageable through both dyadic and triadic learning …
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations such as simplicial complexes and hypergraphs. So far, the choice between simplicial complexes and hypergraphs has often been motivated by technical convenience. Here, using synchronization as an example, we demonstrate that the effects of higher-order interactions are highly representation-dependent. In particular, higher-order interactions typically enhanc…
Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly remov…
We define the topological Dirac equation describing the evolution of a topological wave function on networks or on simplicial complexes. On networks, the topological wave function describes the dynamics of topological signals or cochains, i.e. dynamical signals defined both on nodes and on links. On simplicial complexes the wave function is also defined on higher-dimensional simplices. Therefore the topological wave function satisfies a relaxed condition of locality as it acquires the same value along simplices of dimension larger than zero. The topological Dirac equation defines eigenstates w…
The higher-order interactions of complex systems, such as the brain are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive co…
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable $d$ dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension $d$ for $d\to \infty$ and we poin…
Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate Dirac synchronization, describing locally coupled topological signals defined on the nodes and on the links of a network, and treated using t…
The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunabl…
Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is believed to enhance the stability of identical synchronization. Yet, here we show that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry. This counterintuitive effect holds for generic node dynamics and arbitrary network structure and is, moreover, robust against no…
The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength be…
Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture t…
