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Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks
Louis M. Pecora, Francesco Sorrentino, Aaron M. Hagerstrom, Thomas E. Murphy, Rajarshi Roy
TL;DR
The paper investigates how network symmetries determine synchronized clusters and develops techniques for analyzing their dynamics. It identifies isolated desynchronization, where one cluster loses synchrony while another remains synchronized, and explains this through invariant synchronous states.
Problem
The paper addresses how network symmetries relate to the synchronization or desynchronization of network clusters.
Method
The paper outlines a symmetry-based procedure and represents variational dynamics using matrices associated with irreducible representation multiplicities.
Results
Some clusters can become desynchronized while other clusters remain synchronized because the latter retain identical inputs and an invariant synchronous state.
Takeaways & Limitations
The subgroup decomposition shows that isolated desynchronization is dynamically possible and provides a more general explanation of intertwined desynchronization.
Takeaways & Limitations
Small samples can occasionally miss symmetry cases that are not common in the sample.
Abstract
from arXiv · showhide
Synchronization is of central importance in power distribution, telecommunication, neuronal, and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form, except for in the simplest of networks. In this article, we shed light on the intimate connection between network symmetry and cluster synchronization. We introduce general techniques that use network symmetries to reveal the patterns of synchronized clusters and determine the conditions under which they persist. The connection between symmetry and cluster synchronization is experimentally explored using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony while leaving others synchronized. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behavior of networks ranging from neural to social.
SYMMETRIES, SYNCHRONIZATION CLUSTERS, AND BLOCK DIAGONAL-
The paper uses network symmetries and group representations to identify synchronization clusters and transform the variational equations into independent blocks whose stability determines persistence.
- Symmetry and cluster identification: The symmetry group of the coupling matrix identifies node orbits, which define the network’s synchronization clusters.The procedure extracts the symmetry group, its orbits, permutation representation, and irreducible representations (IRRs).
- Symmetry and cluster identification: The transformation T is constructed from projection-subspace bases for the IRRs and forms an N × N matrix.Singular value decomposition supplies the basis for each IRR subspace, whose row vectors are stacked to construct T.
- Block diagonalization and stability: Applying T block-diagonalizes the variational equations, separating motion within the synchronization manifold from transverse directions.The trivial representation describes motion in the synchronization manifold; all other IRRs correspond to transverse directions.
- Block diagonalization and stability: Each transverse block is analyzed to determine whether nonsynchronous perturbations decay to zero and synchronized clusters remain stable.The blocks are governed by separate variational ordinary differential equations.
- Assumptions: The approach assumes that node dynamics and additional feedback terms commute with the network symmetries.Self-feedback and other terms, such as row sums of the coupling matrix, are allowed when they commute with the symmetry permutations.
SUPPLEMENTARY INFORMATION
The supplementary analysis applies symmetry decomposition to example networks and the Mesa del Sol grid, explaining isolated desynchronization through equal coupling inputs and assessing symmetry statistics in generated graphs.
- Cluster decomposition examples: Nontrivial clusters are identified as groups of nodes sharing network symmetry, with transformation and block-diagonal coupling matrices constructed from those clusters.Examples explicitly list nontrivial node clusters and present the associated transformation and block-diagonal matrices.
- Subgroup decomposition and cluster dynamics: Nodes in one cluster receive identical coupling factors, so another cluster can receive the same input whether the first cluster is synchronized or desynchronized.This equal-input structure makes the second cluster’s synchronous state flow invariant; stability then supports isolated desynchronization.
- Subgroup decomposition and cluster dynamics: Subgroup decomposition generalizes the mechanism to permutations involving several clusters and explains intertwined desynchronization observed in the experiment.The multi-cluster case adds analogous coupling sums to the governing equation.
- Statistics of random graphs: Random and scale-free graph ensembles were generated in 10,000 realizations per graph type to examine symmetry statistics.The scale-free ensembles used power-law exponents γ = 2.5, 3.0, and 3.5.
- Limitations: Small samples can occasionally miss symmetry cases that are not common in the generated graph classes.This is identified as a limitation of using small samples despite the main trends appearing with 100 realizations.
- Small-world networks: Small-world examples required many added edges because adding only a few rarely produced symmetries, making them approach the random-graph cases.The authors therefore do not display their results, although the two systems retain some different symmetries.
- The Mesa del Sol power grid: The 132-node Mesa del Sol grid has 20 nontrivial clusters and 10 subgroups, with about one-third of nodes in synchronizable clusters; its decomposition permits isolated desynchronization.The grid contains many trivial single-node clusters, but the clustered subset supports dynamically possible isolated desynchronization.