5 GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s49>GLYPH<s48> GLYPH<s49> GLYPH<s84> GLYPH<s78> GLYPH<s78> GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s49>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s32> GLYPH<s32>GLYPH<s110> GLYPH<s43> GLYPH<s32> GLYPH<s32>GLYPH<s110> GLYPH<s45> GLYPH<s32> GLYPH<s32>GLYPH<s110> GLYPH<s48> FIG. 5: A plot of T$_{N}$ with N in the log-log coordinate. M (0) = − 1 / 2 , y (0) = 1 / 2. N is the size of the network. T$_{N}$ is the time needed for dynamic balance. The numerical results are obtained by taking the average of 5 × 10 2 simula- tions on the networks with size from 2 × 10 3 to 2 × 10 $^{4}$. FIG. 4: The numerical results are obtained by taking the average of 20 simulations on the network with 10 4 nodes. Ini- tially M (0) = − 1 / 2 , y (0) = 1 / 2, and the network takes a long enough time to get dynamically balanced. The numerical results of the time needed for dynamic balance are given in Fig. 5. More details are given in Appendix A 2. tions are given in Appendix A 1) Since the transition probabilities are not continued at M = 0, we cannot give the analytical results of n $^{+}$, n $^{−}$, n 0 with α , so we turn to the numerical method. The results are shown by Fig. 4. We note when α → 0, n + = n − = n 0 = 1 / 3. And when α → + ∞ , n 0 = 0, which means the neutral opinion disappears, and the ternary network comes back to the binary network. When the network gets dynamically balanced, the dis- tributions of each triangle are as follows, where n + ≃ n − = x and n 0 = y . Besides, i = 1 , 2 , 3 , ..., 10, and l, m, n are the respective numbers of positive, neutral, negative nodes in a$_{i}$ , along with C$_{lmn}$ the corresponding combinatorial number. For example C$_{201}$ = 3. For the time needed for dynamic balance, we expect it to satisfy where N is the size of the network and D ( α ) > 0, being a function of α . If the value of a node being +1, it means the person shows the friendly face in the triad relation, and always tires to keep the relation stable. If the value being -1, it means the person shows the hostile face, and always tries to destroy the stability of the triad relation. While the value being 0, it means the person does not care whether the triad relation is stable or not. So we propose if the As is well known, SOC [6–8] is studied on many com- plex systems, including on the random graph [9], the scale-free network [10–12] and the small-world network [13], etc, which are mainly based on the BTW model [6]. Besides, there are the studies of the SOC on complex net- works based on other models, which we refer the readers to Ref. [14] for details. In this paper, we study the SOC on the ternary social networks based on our model and the evolving rules, along with two modifications com- pared to Sec. II. The first modification is the struc- ture of the network. Because for a completely connected network, any disturbance is globe, with no propagation, so the occurrence of the avalanche will not follow the power-law form either spatially or temporally. Besides, compared to the completely connected network, the reg- ular network or the small-world network [15, 16] is closer to the real structure of the society. So we take the latter forms for study. The second modification is the judge- ment of a triad relation. In order to embody the majority principle, we take the judgement as the gauge invariance or the gauge variance [19–22], which is explained as fol- lows. Under this judgement, the triad relation evolves from one state to the other by changing node’s value. III. SELF-ORGANIZED CRITICALITY p ( − 1 → 0) = 0 , p (1 → 0) = 0 , p (0 → 1) = 1 2 ( a$_{5}$ 3 + a$_{6}$ 3 + a$_{7}$ 3 + 2 a$_{8}$ 3 + 2 a$_{9}$ 3 + a$_{10}$ ) , p (0 → − 1) = 1 2 a$_{5}$ 3 + a$_{6}$ 3 + a$_{7}$ 3 + 2 a$_{8}$ 3 + 2 a$_{9}$ 3 + a$_{10}$ , p (1 → − 1) = 1 2 ( a$_{1}$ + 2 a$_{2}$ 3 + a$_{3}$ 3 + 2 a$_{5}$ 3 + a$_{6}$ 3 + a$_{8}$ 3 ) , p ( − 1 → 1) = 1 2 a$_{2}$ 3 + 2 a$_{3}$ 3 + a$_{4}$ + a$_{6}$ 3 + 2 a$_{7}$ 3 + a$_{9}$ 3 . (17) a$_{i}$ ≃ C$_{lmn}$x l + $^{n}$y$^{m}$, (18) T$_{N}$ ∼ N D ( α $^{)}$, (19)