11 where j = 5 , 6 , 7 , 8 , 9 , 10. And the densities of each tri- angle attached to a negative node are and ( M → 0) The densities of each triangle attached to a positive node are where k = 2 , 3 , 4 , 6 , 7 , 9. n − a$_{k}$ ≃ c$_{1}$x 2 + 2 c$_{2}$x 2 + 2 c$_{3}$xy + c$_{4}$y 2 (2 x + y ) 2 , (A19) p ( − 1 → 0) = 0 , p (1 → 0) = 0 , p (0 → 1) = 2 x $^{2}$y + 2 xy 2 + y 3 2 , p (0 → − 1) = 2 x $^{2}$y + 2 xy 2 + y 3 2 , p (1 → − 1) = 2 x 3 + 2 x $^{2}$y + xy 2 2 , p ( − 1 → 1) = 2 x 3 + 2 x $^{2}$y + xy 2 2 . (A16) n + a$_{i}$ ≃ c$_{1}$x 2 + 2 c$_{2}$x 2 + 2 c$_{3}$xy + c$_{4}$y 2 (2 x + y ) 2 , (A17) n 0 a$_{j}$ ≃ c$_{1}$x 2 + 2 c$_{2}$x 2 + 2 c$_{3}$xy + c$_{4}$y 2 (2 x + y ) 2 , (A18) Because the transition probabilities (see Eq. (16) and Eq. (17)) are not continued at M = 0, we cannot obtain the analytical results for the distributions of n $^{+}$, n − and n 0 with α , so we turn to the numerical method, as shown by Fig. 4. We note when α → + ∞ , the neutral opinion disappears, and the ternary network comes back to the binary network. That can be explained as follows. In the triad relation, the more people try to keep it stable, i.e., to show the same opinion with the others, the less likely the neutral opinion will exist. where i = 1 , 2 , 3 , 5 , 6 , 8, and ∑ 4 j $_{=1}$c$_{j}$ = 1 , c$_{j}$ = 0 , 1. That is, only one of the coefficient c$_{j}$ is nonzero. In a similar way, the densities of each triangle attached to a neutral node are 338 , 84 (2004). [1] T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 72 , 036121 (2005). [2] F. Heider, Psychol. Rev. 51 , 358 (1944); F. Heider, J. Psychol. 21 , 107 (1946); F. Heider, The Psychology of Interpersonal Relations (Wiley, New York, 1958). [3] S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications (Cambridge University Press, New York, 1994). [12] D. S. Lee, K. I. Goh, B. Kahng, and D. Kim, J. Korean Phys. Soc. 44 , 633 (2004). [13] L. de Arcangelis and H. Herrmann, Physca A 308 , 545 (2002). [14] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. 80 , 1275 (2008). [15] D. J. Watts and S. H. Strogatz, Nature 393 , 440 (1998). [16] M. E. J. Newman and D. J. Watts, Phys. lett. A 263 , 341 (1999). [17] R. J. Glauber, J. Math. Phys. 4 , 294 (1963). [4] F. Radicchi, D. Vilone, and H. Meyer-Ortmanns, Phys. Rev. E 75 , 021118 (2007). [18] J. Y. Zhu and Z. R. Yang, Phys. Rev. E 59 , 1551 (1999); H. Zhu, J. Y. Zhu, and Y. Zhou, Phys. Rev. E 66 , 036106 (2002). [5] F. Radicchi, D. Vilone, S. Yoon, and H. Meyer- Ortmanns, Phys. Rev. E 75 , 026106 (2007). [6] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59 , 381 (1987). [7] P. Bak, How Nature Works (Springer-Verlag, New York, 1996). [8] H. J. Jensen, Self Organized Criticality (Cambridge Uni- versity Press, 1998). [9] E. Bonabeau, J. Phys. Soc. Jpn. 64 , 327 (1995). [10] K. I. Goh, D. S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91 , 148701 (2003). [11] D. S. Lee, K. I. Goh, B. Kahng, and D. Kim, Physica A [19] J. Z. Chen and J. Y. Zhu, Int. J. Mod. Phys. A 23 , 3891 (2008). [20] C. Rovelli and L. Smolin, Phys. Rev. D 52 , 5743 (1995). [21] J. C. Baez, Adv. Math. 117 , 253 (1996). [22] M. H. Ansari and L. Smolin, Class. Quant. Grav. 25 , 095016 (2008).