Lifetime distributions for adjacency relationships in a Vicsek model

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(τ) of lifetime τ of the edges. We find that the shape of P(τ) 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.