Geometric unfolding of synchronization dynamics on networks

Area of Science:

• Complex Systems
• Network Science
• Dynamical Systems

Background:

• Understanding synchronized states in coupled oscillator populations is crucial in various scientific fields.
• Heterogeneity in oscillator frequencies and network topology complicates analysis of synchronization dynamics.

Purpose of the Study:

• To develop a new analytical framework for characterizing the synchronized state in network-coupled heterogeneous oscillators.
• To demonstrate the convergence properties and error bounds of the proposed method.
• To derive a practical local approximation for synchronized states.

Main Methods:

• Linearized dynamics of coupled heterogeneous oscillators.
• Geometric series expansion of the steady-state solution.
• Analysis of network adjacency matrices and eigenvalues.
• Truncation of the spatial series for local approximation.

Main Results:

• The steady-state solution can be expressed as a geometric series representing spatial scales.
• The series converges for diverse frequency distributions and network types (undirected/directed) if the adjacency matrix is primitive.
• Truncation error is geometrically linked to the second largest eigenvalue of the normalized adjacency matrix.
• A local approximation effectively captures synchronization by considering nearest-neighbor interactions.

Conclusions:

• The geometric series expansion provides a robust and convergent method for analyzing synchronization in complex oscillator networks.
• The derived error bound offers insights into the convergence rate, analogous to random walk dynamics.
• The local approximation demonstrates practical utility for simplifying analysis in large-scale networks.