Lifetime distributions for adjacency relationships in a Vicsek model

Area of Science:

• Statistical Physics
• Complex Systems
• Agent-Based Modeling

Background:

• The Vicsek model is a fundamental framework for studying self-propelled particles and emergent collective motion.
• Understanding the statistical properties of particle interactions is crucial for characterizing emergent behaviors in dynamic systems.

Purpose of the Study:

• To investigate the statistical properties of adjacency relationships in a 2D Vicsek model.
• To determine how interaction parameters influence the lifetime distributions of particle-to-particle edges.

Main Methods:

• Defined adjacent edges using Delaunay triangulation and Euclidean distance at each time step.
• Calculated cumulative distributions P(τ) for the lifetime τ of these edges.
• Analyzed the transition in distribution shape as a function of the Vicsek model's interaction radius.

Main Results:

• The lifetime distribution of adjacency edges transitions from exponential to power-law.
• This transition is dependent on the interaction radius parameter of the Vicsek model.
• The power-law emergence is discussed in the context of first passage time problems for fractional Brownian motion.

Conclusions:

• The interaction radius is a critical parameter governing the statistical nature of particle adjacency in the Vicsek model.
• The findings suggest a connection between local interaction rules and global emergent statistical properties.
• The study provides a framework for analyzing dynamic networks in agent-based models.