• Corpus ID: 244478626

Connecting Hodge and Sakaguchi-Kuramoto: a mathematical framework for coupled oscillators on simplicial complexes

  title={Connecting Hodge and Sakaguchi-Kuramoto: a mathematical framework for coupled oscillators on simplicial complexes},
  author={Alexis Arnaudon and Robert L. Peach and Giovanni Petri and Paul Expert},
We formulate a general Kuramoto model on weighted simplicial complexes where phases oscillators are supported on simplices of any order k . Crucially, we introduce linear and non-linear frustration terms that are independent of the orientation of the k +1 simplices, providing a natural generalization of the Sakaguchi-Kuramoto model. In turn, this provides a generalized formulation of the Kuramoto higher-order parameter as a potential function to write the dynamics as a gradient flow. With a… 

Figures from this paper

Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes.

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional "simplices") and how

Do higher-order interactions promote synchronization?

Understanding how nonpairwise interactions alter dynamical processes in networks is of funda-mental importance to the characterization and control of many coupled systems. Recent discoveries of

Dirac synchronization is rhythmic and explosive

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

Multistability and Paradoxes in Lossy Oscillator Networks

It is shown that loop flows and Dissipation can increase the system’s transfer capacity, and that dissipation can promote multistability, three paradoxical behaviors of lossy networked systems, to be contrasted with the behavior lossless systems.

Synchronization induced by directed higher-order interactions

Non-reciprocal interactions play a crucial role in many social and biological complex systems. While directionality has been thoroughly accounted for in networks with pairwise interactions, its

Multistability and anomalies in oscillator models of lossy power grids

The analysis of dissipatively coupled oscillators is challenging and highly relevant in power grids. Standard mathematical methods are not applicable, due to the lack of network symmetry induced by

Dynamics on higher-order networks: a review

A variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation, are studied.

Topological synchronization: explosive transition and rhythmic phase

This paper presents a meta-analyses of the response of the immune system to Tournais' inequality and shows clear patterns in response to different types of immune systems.



Explosive higher-order Kuramoto dynamics on simplicial complexes

The higher-order Kuramoto model is formulated which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on and can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Higher-order simplicial synchronization of coupled topological signals

This work investigates a framework of coupled topological signals where oscillators are defined both on the nodes and the links of a network, showing that this leads to new topologically induced explosive transitions.

Control Using Higher Order Laplacians in Network Topologies

This paper establishes the proper notation and precise inte rpre ation for Laplacian flows on simplicial complexes. In particular, we have shown how to interpret these flows as ti me-varying discrete

Consensus on simplicial complexes: Results on stability and synchronization.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes.

This analysis of the collective dynamics of a large ensembles of dynamical units with nonpairwise interactions, namely coupled phase oscillators with three-way interactions, sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplified interactions play in storing information.

The Kuramoto model in complex networks

Dynamical systems on hypergraphs

Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs.

Simplicial models of social contagion

Higher-order social interactions, the effects of groups, are included in their model of social contagion, enabling insight into why critical masses are required to initiate social changes and contributing to the understanding of higher-order interactions in complex systems.

Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching

These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.

The Kuramoto Model on Oriented and Signed Graphs

This work uses a generalized Kuramoto model with oriented, weighted and signed interactions to investigate the simplest possible oriented networks, namely acyclic oriented networks and oriented cycles, and proves that if it exists, a stable synchronous state is unique.