content_image/339322.jpg
<img src='content_image/339322.jpg'>

entropies as a friend of node u . We repe at the adding execution until all the candidates are chosen or the energy limit (Eq. (2)) is satisfied. 

In the evolutionary model, we have to record all friends of each node and therefore a system of size $^{L}$× L with L = 10000 is too larg e to simulate. So we simulate the evolutionary model on a toroidal lattice of size 100 × 100. W e assume that the energy scales linearly with distance as suggested by Eq. (2). Thus, when reduci ng L from 10,000 to 100 (factor of 100) we expect the corresponding energy to be reduced from order of 10 5 to order of 10$^{3}$. We therefore study the EM model of L = 100 with w = 1086 ( f = 5 0). 

In order to find the optimal distributio n of the distances, wefirst employ the optimization model described by Eqs. (2)-(4) to analyze the a bove case with the system size 100 × 100 and w ≃ 10$^{3}$. We find that the maximum entropy is 7. 18 and the corresponding α is α = 0 . 95 ± 0 . 05 (see Fig. 3a, b). Next we simulate the evolutionar y model of size of 100 × 100 and w ≃ 10$^{3}$. After long 

