Synchronization and equitable partitions in weighted networks.

Area of Science:

• Dynamical Systems and Network Theory
• Mathematical Biology
• Graph Theory

Background:

• Coupled cell network formalism traditionally uses nonnegative integer connection values.
• Existing literature lacks a unified framework for weighted networks in this context.

Purpose of the Study:

• To demonstrate the natural extension of the coupled cell network formalism to weighted networks.
• To establish a correspondence between existing concepts and those in weighted network analysis.
• To link two distinct but related strands of network literature.

Main Methods:

• Applying central concepts and results of the existing formalism to weighted networks.
• Analyzing synchrony subspaces and balanced relations in weighted coupled cell networks.
• Establishing equivalences between synchrony subspaces/balanced relations and cluster of synchrony/equitable partitions.

Main Results:

• The coupled cell network formalism effectively accommodates weighted networks.
• Key results concerning synchrony subspaces and balanced relations are generalized to weighted networks.
• A direct correspondence is shown between synchrony subspaces and cluster of synchrony, and balanced relations and equitable partitions.

Conclusions:

• The generalized formalism provides a unified framework for analyzing both unweighted and weighted coupled cell networks.
• This work bridges the gap between different theoretical approaches in network science.
• The findings are applicable to a broader range of network dynamics and systems.