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 $P\left(\tau \right)$ of lifetime $\tau$ of the edges. We find that the shape of $P\left(\tau \right)$ 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.

Figure 1

Images of particles' motion and adjacency relationship for $N=200$ and $\eta =0.2$ at $t=1100$ in the circular reflection boundary with diameter $L=1$. The direction of the particles is shown by arrows. Lines between particles show the adjacency networks, defined by (a) the Delaunay triangulation and (b) the Euclidean distance $d<0.05$. Left column: $R_0=0$. Particles move randomly because they do not interact. Middle column: $R_0=0.05$. Particles' directions are aligned locally. Right column: $R_0=0.1$. All particles form a single cluster and move along the boundary.

Figure 2

Cumulative distributions $P\left(\tau \right)$ for $R_0=0$ obtained from (a) Delaunay and (b) Euclidean edges in a semilogarithmic scale. For small $\eta$, exponential decay is observed.

Figure 3

Cumulative distributions $P\left(\tau \right)$ for (a) Delaunay and (b) Euclidean edges. Each panel is shown in a double logarithmic scale. The slope of guideline in each panel is obtained by fittings for $R_0=0.1$ and $\eta =0.2$. Left column: $R_0$ dependence of $P\left(\tau \right)$ where $\eta =0.2$; $P\left(\tau \right)$ changes from an exponential to a power-law distributions with an increase in $R_0$. Middle column: $\eta$ dependence of $P\left(\tau \right)$ where $R_0=0.1$. Right column: $R_0$ dependence of CV values.

Figure 4

(a) $\eta$ dependence of $P\left(\tau \right)$ obtained from Delaunay edges without boundaries where $R_0=0.2$. (b) $R_f$ dependence of $P\left(\tau \right)$ for Delaunay edges where $R_0=0.1$ and $\eta =0.2$.

Figure 5

Behaviors of $P\left(\tau \right)$ as functions of $R_0$ and $\eta$. A: exponential; B: power law; C: crossover between A and B; D: $P\left(\tau \right)$ becomes a distribution with a longer tail than the power law such as Fig. 4. The vertical dashed line at $R_0\simeq 0.05$ represents the point around which $\mathrm{CV}$ changes considerably.

Figure 6

Typical time series of (a) length $l\left(t\right)$ of Voronoi line and (b) distance $r\left(t\right)$ between two particles. Dashed lines indicate $l\left(t\right)=0$ and $r\left(t\right)=0.05$, which show the thresholds for the first return time. Left column: $R_0=0.01$ and $\eta =0.2$ in case A of Fig. 5. Middle column: $R_0=0.1$ and $\eta =0.2$ in case B. Right column: probability distributions of Hurst exponent $H$ for the time series at $R_0=0.1$ and $\eta =0.2$. Dashed lines indicate the peak values.