Perfect cycles in the synchronous Heider dynamics in complete network.

Area of Science:

• Computational Social Science
• Network Science
• Complex Systems

Background:

• Heider balance theory describes social balance in triads.
• Cellular automata provide a framework for simulating complex system dynamics.
• Understanding attractors in dynamical systems is crucial for predicting long-term behavior.

Purpose of the Study:

• To investigate a cellular automaton model for achieving Heider balance in fully connected networks.
• To identify and characterize a specific class of limit cycles, termed 'perfect' limit cycles.
• To explore the symmetry properties of these perfect limit cycles and their connection to network structure.

Main Methods:

• Development of a deterministic, synchronous, global update rule for the cellular automaton.
• Analysis of the attractor spectrum, focusing on fixed points and limit cycles.
• Characterization of 'perfect' limit cycles based on energy spectrum preservation and Hamming distance properties.
• Investigation of symmetries in perfect cycle trajectories related to network and energy function symmetries.

Main Results:

• The cellular automaton exhibits a rich spectrum of attractors, including fixed points and limit cycles, whose properties vary with system size.
• A novel class of 'perfect' limit cycles was identified, characterized by constant Hamming distances between consecutive states and across k-step separations.
• Perfect limit cycles display high symmetry in the configuration space.
• The observed symmetries are linked to the permutation symmetry of network vertices and local symmetries in the energy function.

Conclusions:

• The study reveals that perfect limit cycles represent a highly symmetric and structured form of dynamics in the Heider balance model.
• The findings suggest a deep connection between the topological symmetries of the network and the emergent dynamical behaviors.
• The identified perfect limit cycles offer insights into the fundamental principles governing balance and frustration in social networks.