# Graph partitions and cluster synchronization in networks of oscillators

arXiv: https://arxiv.org/abs/1608.04283 · HTML: https://mlsanity.com/paper/1608.04283/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2016.

## Built on by (most cited first, 6 of 6)

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- [A nonlinear diffusion method for semi-supervised learning on hypergraphs](https://mlsanity.com/paper/2103.14867/) (2021)
- [KuraNet: Systems of Coupled Oscillators that Learn to Synchronize](https://mlsanity.com/paper/2105.02838/) (2021)
- [Blind Extraction of Equitable Partitions from Graph Signals](https://mlsanity.com/paper/2203.05407/) (2022)
- [On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs](https://mlsanity.com/paper/2304.07203/) (2023)
- [Data-Topology Compatibility Enables Personalization in Decentralized Learning](https://mlsanity.com/paper/2610.15126/) (2026)

## Abstract (arXiv)

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges, and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.