Graphical notation reveals topological stability criteria for collective dynamics in complex networks

Abstract

We propose a graphical notation by which certain spectral properties of complex systems can be rewritten concisely and interpreted topologically. Applying this notation to analyze the stability of a class of networks of coupled dynamical units, we reveal stability criteria on all scales. In particular, we show that in systems such as the Kuramoto model the Coates graph of the Jacobian matrix must contain a spanning tree of positive elements for the system to be locally stable.

5 pages, 3 figures

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Cited by in corpus (12)

• The Kuramoto model in complex networks
• Explosive transitions in complex networks' structure and dynamics: percolation and synchronization
• Dynamical Systems on Networks: A Tutorial
• Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models on Single-Loop Networks
• Kosambi-Cartan-Chern (KCC) theory for higher order dynamical systems
• Effect of molecular rotation on enantioseparation
• Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs
• Exploring the adaptive voter model dynamics with a mathematical triple jump
• Moment-Closure Approximations for Discrete Adaptive Networks
• Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
• Efficient hierarchical analysis of the stability of a network through dimensional reduction of its influence topology
• A note on Graphical Notation Reveals Topological Stability Criteria for Collective Dynamics in Complex Network