Acknowledgments Appendix A: Distributions of each triad relation and the time needed for dynamic balance 1. The node model 9 a$_{1}$ = 1 8 , a$_{2}$ = 3 8 , a$_{3}$ = 3 8 , a$_{4}$ = 1 8 . (A4) a$_{i}$ = C j $_{3}$ρ $^{j}$(1 − ρ ) 3 − $^{j}$, M = n + − n − = 2 ρ − 1 , ρ = (1 + M ) / 2 , ρ ( ∞ ) = 1 2 , (A5) dn + dt = p ( − 1 → 1) − p (1 → − 1) , dn − dt = p (1 → − 1) − p ( − 1 → 1) , dM dt = dn + dt − dn − dt = 2[ p ( − 1 → 1) − p (1 → − 1)] . (A6) da$_{1}$ dt = da$_{2}$ dt = da$_{3}$ dt = da$_{4}$ dt = 0 , (A2) da$_{1}$ dt = p ( − 1 → 1) n − a$_{2}$ − p (1 → − 1) n + $_{a$_{1}$}$, da$_{2}$ dt = p (1 → − 1) n + a$_{1}$ + p ( − 1 → 1) n − a$_{3}$ − p (1 → − 1) n + a$_{2}$ − p ( − 1 → 1) n − $_{a$_{2}$}$, da$_{3}$ dt = p (1 → − 1) n + a$_{2}$ + p ( − 1 → 1) n − a$_{4}$ − p (1 → − 1) n + a$_{3}$ − p ( − 1 → 1) n − $_{a$_{3}$}$, da$_{4}$ dt = p (1 → − 1) n + a$_{3}$ − p ( − 1 → 1) n − $_{a$_{4}$}$. (A1) p ( − 1 → 1) = p (1 → − 1) . (A3) dM dt = 2 [ ( 1 + M 2 ) $^{2}$( 1 − M 2 ) e 3 α/ (1 − M ) e 3 α/ (1 − M ) + e − α/ (1+ M ) + 2 ( 1 + M 2 )( 1 − M 2 ) 2 e − α/ (1 − M ) e − α/ (1 − M ) + e − α/ (1+ M ) + ( 1 − M 2 ) 3 e − α/ (1 − M ) e − α/ (1 − M ) + e 3 α/ (1+ M ) − ( 1 + M 2 ) 3 e − α/ (1+ M ) e − α/ (1+ M ) + e 3 α/ (1 − M ) − 2 ( 1 + M 2 ) $^{2}$( 1 − M 2 ) e − α/ (1+ M ) e − α/ (1+ M ) + e − α/ (1 − M ) − ( 1 + M 2 )( 1 − M 2 ) 2 e 3 α/ (1+ M ) e 3 α/ (1+ M ) + e − α/ (1 − M ) ] (A7) M ( t ) ∼ t − 1 /C ( α $^{)}$, (A8) | M ( T$_{N}$ ) | ∼ O ( 1 N ) . (A9) From Eqs. (A1), (A2) and (A3), along with the normal- ization ∑ 4 i $_{=1}$a$_{i}$ = 1, we obtain We note the distributions of each triad relation indepen- dent of the parameter α , and n + = n − in the dynamically balanced state. Defining at any time the density of positive nodes to be ρ , we obtain We thank H. Zhu for helpful discussions. The work was supported by the National Natural Science Foundation of China (No. 10875012). For the node model, each node has two spin values, +1 and -1. We define one step of time as executing N spin flipping successively at random [1]. Because each spin flipping is linked up with ( N − 1)( N − 2) / 2 variations of the triangles, the variations of the densities of each triad relation with time are as follows, where j is the number of positive nodes in a$_{i}$ and C j 3 is the combinatorial number. We have the new equations as follows, When the network gets dynamically balanced, we have and Substitute Eqs. (8), (9) and (A5) into Eq. (A6), we obtain For Eq. (A7), when M ( t ) ∼ 0, after some calculations, we obtain the approximate result, For the dynamic balance of the network, we propose where C ( α ) > 0, being a function of α .