# Lifetime distributions for adjacency relationships in a Vicsek model

## TL;DR

This paper studies the lifetime distributions of adjacency relationships in a Vicsek model, revealing a transition from exponential to power-law distributions based on interaction radius, with implications for understanding collective motion.

## Contribution

It introduces a method to analyze adjacency lifetime distributions in a Vicsek model and links the distribution shape to the interaction radius, providing new insights into collective dynamics.

## Key findings

- Distribution shape changes from exponential to power law with interaction radius
- Power-law emergence explained via fractional Brownian motion
- Provides a statistical framework for adjacency analysis in collective motion

## Abstract

We investigate the statistical properties of adjacency relationships in a two-dimensional Vicsek model. We define adjacent edges for all particles at every time step by (a) Delaunay triangulation and (b) Euclidean distance, and obtain cumulative distributions $ P(\tau) $ of lifetime $ \tau $ of the edges. We find that the shape of $ P(\tau) $ changes from an exponential to a power law depending on the interaction radius, which is a parameter of the Vicsek model. We discuss the emergence of the power-law distribution from the viewpoint of first passage time problem for a fractional Brownian motion.