10 2. The generalized model T$_{N}$ ∼ N C ( α $^{)}$. (A10) da$_{1}$ dt = p ( − 1 → 1) n − a 2 + p (0 → 1) n 0 a 5 − [ p (1 → 0) + p (1 → − 1)] n + a $_{1}$, da$_{2}$ dt = p (1 → − 1) n + a 1 + p ( − 1 → 1) n − a 3 + p (0 → − 1) n 0 a 5 + p (0 → 1) n 0 a 6 − [ p (1 → 0) + p (1 → − 1)] n + a 2 − [ p ( − 1 → 0) + p ( − 1 → 1)] n − a $_{2}$, . da$_{9}$ dt = p (1 → 0) n + a 6 + p ( − 1 → 0) n − a 7 + p (1 → − 1) n + a 8 + p (0 → − 1) n 0 a 10 − [ p (0 → − 1) + p (0 → 1)] n 0 a 9 − [ p ( − 1 → 0) + p ( − 1 → 1)] n − a $_{9}$, da$_{10}$ dt = p (1 → 0) n + a 8 + p ( − 1 → 0) n − a 9 − [ p (0 → − 1) + p (0 → 1)] n 0 a $_{10}$. (A11) da$_{i}$ dt = 0 , (A12) a$_{i}$ ≃ C$_{lmn}$x l + $^{n}$y$^{m}$, (A14) 10 ∑ i =1 a$_{i}$ = 1 . (A13) p ( − 1 → 0) ≃ x 3 + x 3 + 2 x $^{2}$y 1 + e α + e − α + 2 x $^{2}$y 1 + e α + e − α + xy 2 3 , p ( − 1 → 1) ≃ x 3 + 2 x $^{2}$ye α 1 + e α + e − α + 2 x $^{2}$ye − α 1 + e α + e − α + xy 2 3 , p (0 → − 1) ≃ x $^{2}$y + 2 xy $^{2}$e − α 1 + e α + e − α + 2 xy $^{2}$e α 1 + e α + e − α + y 3 3 , p (0 → 1) ≃ x $^{2}$y + 2 xy $^{2}$e − α 1 + e α + e − α + 2 xy $^{2}$e α 1 + e α + e − α + y 3 3 , p (1 → − 1) ≃ x 3 + 2 x $^{2}$ye − α 1 + e α + e − α + 2 x $^{2}$ye α 1 + e α + e − α + xy 2 3 , p (1 → 0) ≃ x 3 + x 3 + 2 x $^{2}$y 1 + e α + e − α + 2 x $^{2}$y 1 + e α + e − α + xy 2 3 , (A15) So the time needed for dynamic balance is network will reach the dynamically balanced state. In the simulations, both for the binary and the ternary cases, if the final value σ ′ i results in the denominator of β (see Eq. (3)) being zero, we propose it happens with probability zero, while the other values happen with probability one (see Eq. (17) for example). In this man- ner, no matter what M (0) (or M (0) and ρ (0)) is, the When each node has three spin values, the variations of the densities of each triad relation with time are as follows, When the network gets dynamically balanced, it satisfies we expect n + ≃ n − and M ( ∞ ) ≃ 0 in the dynamically balanced state. Defining n + ≃ n − = x and n 0 = y in the dynamically balanced state, the densities of each type of the triangles are where i = 1 , 2 , 3 , ..., 10, along with the normalization where l, m and n are the numbers of positive, neutral and negative nodes in a$_{i}$ respectively. C$_{l}$mn is the corre- sponding combinatorial number. For example C$_{201}$ = 3. Because near M = 0, there are fluctuations. The spin transition probabilities are ( M = 0) Although Eqs. (A11) are very hard to solve, from ap- pendix A 1 and the symmetric spin transition mechanism,