# Forget Partitions: Cluster Synchronization in Directed Networks Generate Hierarchies

@inproceedings{Brady2021ForgetPC, title={Forget Partitions: Cluster Synchronization in Directed Networks Generate Hierarchies}, author={Fiona M. Brady and Yuanzhao Zhang and Adilson E. Motter}, year={2021} }

We present a scalable approach for simplifying the stability analysis of cluster synchronization patterns on directed networks. When a network has directional couplings, decomposition of the coupling matrix into independent blocks (which in turn decouples the variational equation) is no longer adequate to reveal the full relations among perturbation modes. Instead, it is often necessary to introduce directional dependencies among the blocks and establish hierarchies among perturbation modes…

## 4 Citations

Analyzing states beyond full synchronization on hypergraphs requires methods beyond projected networks

- Computer Science
- 2021

Considering the full hypergraph is required for analyzing the most general types of dynamics on hypergraphs and it is shown that a formulation based on node clusters and the corresponding edge clusters induced by the node partitioning enables the analysis of admissible patterns of cluster synchronization and their effective dynamics.

Do higher-order interactions promote synchronization?

- MathematicsArXiv
- 2022

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…

Comment on "Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks"

- Computer ScienceArXiv
- 2021

Although the results in Panahi et al. are technically valid, their interpretations are misleading and akin to claiming a community detection algorithm to be a failure because it does not find any meaningful communities in Erdős-Rényi networks.

Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions

- Computer Science
- 2021

A unified framework is established to simplify the stability analysis of cluster synchronization patterns for a wide range of generalized networks, including hypergraphs, multilayer networks, and temporal networks and demonstrates the existence of chimera states that emerge exclusively in the presence of higher-order interactions.

## References

SHOWING 1-10 OF 76 REFERENCES

One-way dependent clusters and stability of cluster synchronization in directed networks

- Computer ScienceNature communications
- 2021

The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form and decompose the original problem into subproblems of the lowest dimension, which allows to immediately detect inter-dependencies among clusters.

Symmetry-Independent Stability Analysis of Synchronization Patterns

- Computer ScienceSIAM Rev.
- 2020

A generalization of the MSF formalism is established that can characterize the stability of any cluster synchronization pattern, even when the oscillators and/or their interactions are nonidentical, and leads to an algorithm that is error-tolerant and orders of magnitude faster than existing symmetry-based algorithms.

Maximum performance at minimum cost in network synchronization

- Computer Science, Mathematics
- 2006

Synchronization and equitable partitions in weighted networks.

- MathematicsChaos
- 2018

The work presented in this paper has two purposes. One is to expose that the coupled cell network formalism of Golubitsky, Stewart, and collaborators accommodates in a natural way the weighted…

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

- MathematicsJ. Nonlinear Sci.
- 2011

Focussing on transitive networks that have only one type of cell (identical cell networks), this work addresses three questions relating the network structure to dynamics, and investigates how the dynamics of coupled cell networks with different structures and numbers of cells can be related.

Cluster and group synchronization in delay-coupled networks.

- Mathematics, PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

This work investigates the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics and finds that the master stability function shows a discrete rotational symmetry depending on the number of groups.

Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

- Mathematics, Computer ScienceIEEE Transactions on Control of Network Systems
- 2020

Quantitative conditions on the network weights, cluster configuration, and oscillators’ natural frequency that ensure the asymptotic stability of the cluster synchronization manifold are derived, showing that cluster synchronization is stable when the intracluster coupling is sufficiently stronger than the intercluster coupling.

Characterization and Computation of Partial Synchronization Manifolds for Diffusive Delay-Coupled Systems

- MathematicsSIAM J. Appl. Dyn. Syst.
- 2016

This work presents equivalent existence criteria for partial synchronization manifolds in terms of invariant spaces, the block-structure of a reordered adjacency matrix, and the solvability of a Sylvester equation.

Topological Control of Synchronization Patterns: Trading Symmetry for Stability.

- Computer SciencePhysical review letters
- 2019

It is shown that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry, which holds for generic node dynamics and arbitrary network structure.

Nonlinear dynamics of networks: the groupoid formalism

- Mathematics
- 2006

A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that preserve
the network topology, has existed for some time.…