Figure 3. Boundary effect.

(a1), (b1), (c1), Behavior of 〈w(t)〉_b versus b from direct simulation (blue squares) and predicted by Eq. (9) (solid black lines) for v = 0, 0.5 and 3.0, respectively. Note that v ≈ 0.5 is the optimal value of the velocity that minimizes n_ex in Fig. 1, and v = 3.0 represents a relatively high velocity value for the system of size L = 10 and communication radius a = 1. For (a1), we have η(v = 0) = 0. Since Eq. (9) indicates a smooth dependence of 〈w(t)〉_b on b and since a detailed interdependence between η(v) and v is unknown, we find η(v) = 15 best fits the case in (b1). For (c1), we have η(v = 3.0) → η(v → ∞) = 50. The red dashed line indicates the position of the peak in the curve of 〈w(t)〉_b versus b. (a2), (b2), (c2), Positions of all extreme events occurred in square domain for v = 0, 0.5, and 3.0, respectively. (a3), (b3), (c3), Positions of all extreme events occurred in circular domain for v = 0, 0.5, and 3.0, respectively. (a4), (b4), (c4), Positions of all extreme events occurred in stadium domain for v = 0, 0.5, and 3.0, respectively. All results are obtained from 20 independent statistical realizations.