Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model

Area of Science:

• Complex Systems
• Network Science
• Nonlinear Dynamics

Background:

• Coupled oscillator models, like the Kuramoto model, primarily focus on phase synchronized states.
• Global constraints, such as volume conservation in fluid dynamics, can prevent synchronization, necessitating the study of phase balanced states.
• Previous research established stable balanced states on circulant graphs, but noncirculant graphs remain underexplored.

Purpose of the Study:

• To identify noncirculant graphs that support stable phase balanced states.
• To characterize the properties of these balanced states on novel network structures.
• To develop a framework for understanding oscillator networks under global constraints.

Main Methods:

• Derivation of rules for constructing noncirculant, planar graphs that allow for balanced states from a base cycle graph.
• Identification of different classes of small planar networks supporting balanced states.
• Analysis of basin stability variance and introduction of the balancing ratio as a new order parameter.

Main Results:

• Construction rules for noncirculant planar graphs admitting balanced states were derived.
• Different classes of small planar networks supporting balanced states were identified.
• The variance in basin stability was found to scale linearly with graph size for these networks.

Conclusions:

• This work provides an analytical description of noncirculant graphs that support stable, phase balanced states.
• The findings offer insights into the topological requirements for oscillator networks operating under global constraints.
• The introduction of the balancing ratio provides a new tool for classifying balanced states.