Universality class of triad dynamics on a triangular lattice.

Area of Science:

• Statistical Physics
• Network Science
• Social Dynamics

Background:

• Social balance theory, particularly triad dynamics, has been explored on all-to-all networks.
• Generalizing triad dynamics to regular lattice topologies is crucial for understanding complex social structures.

Purpose of the Study:

• To generalize triad dynamics to a 2D triangular lattice and investigate its phase transitions.
• To analyze the impact of a propensity parameter (p) on network structure and dynamics.
• To determine critical exponents and fractal dimensions associated with the observed phase transitions.

Main Methods:

• Simulating triad dynamics on a 2D triangular lattice with varying propensity parameter p.
• Employing finite-size scaling analysis to determine static and dynamic critical exponents.
• Investigating the formation and properties of negative link clusters.

Main Results:

• A phase transition is identified at a critical propensity p(c), separating distinct absorbing states.
• For p <= p(c), an infinite cluster of negative links forms, and relaxation time scales power-like with system size.
• For p > p(c), relaxation time scales logarithmically with system size.
• Static and dynamic critical exponents were determined, satisfying hyperscaling relations.
• The fractal dimension was calculated and also satisfied a hyperscaling relation.
• The transition belongs to a universality class with distinct critical exponents.
• Generalization to four-cycle dynamics on a square lattice showed similar transitions but violated standard scaling relations.

Conclusions:

• The generalized triad dynamics on a triangular lattice exhibits a rich phase structure characterized by critical phenomena.
• The propensity parameter p critically influences the network's absorbing state and dynamics.
• The study establishes a new universality class for triad dynamics on lattices.
• The violation of scaling relations in the four-cycle dynamics suggests complex behavior in higher-order structures.