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(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.