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P ( t ) is the distribution function. (a) q = 0 . 05, δ = 1 . 0, and the exponent of the power-law distribution is -2.56. (b) q = 0 . 05, δ = 4 . 0, and the exponent of the power-law distribution is -2.22. FIG. 11: A plot of P ( s ) versus s on the small-world network considering the distance effect. q is the edge adding prob- ability. s is the size of avalanche. P ( s ) is the distribution function. (a) q = 0 . 05, δ = 1 . 0, and the exponent of the power-law distribution is -1.85. (b) q = 0 . 05, δ = 4 . 0, and the exponent of the power-law distribution is -1.69. IV. CONCLUSIONS work, when considering the factor of distance, the small- world effect on the triangles far away compared to the regular network is weakened, which may be easier for the spread. Besides, the number of local triangles are in- creased. So we expect the size of avalanche to increase, and the exponent of the size distribution to increase ac- cordingly. For the small-world network, when considering the distance effect, it is constructed as follows. From a regular network, as shown by Fig. 6, for each node in the network, it sends out an edge with probability q and attaches to another node in the network. The probability of attachment between nodes i and j is We have studied the social networks based on our model and obtained meaningful results. First, in Sec. II of this paper, based on Ref. [1], we map the edge re- lations to the node relations with some relaxation, along with introducing the neutral relation. So the social net- works change from the binary networks into the ternary ones. We suppose the change of each triad relation de- pending both on itself and the surroundings. So based on our model, when the social networks get dynamically bal- anced, we obtain the distributions of each triad relation and the time needed for dynamic balance. Besides, we find under the extreme condition α → + ∞ , the ternary networks come back to the binary ones. Second, in Sec. III of this paper, we study the SOC on the ternary social networks based on our model. Because we suppose, if one triad relation changes, it will affect other triad relations associating with it. The effect may be big or small, de- pending on many internal factors. While compared with Sec. II of this paper, we make two modifications for our node model as follows. One is the structure of the social networks, which takes the form of regular or small-world network. The other is the judgement of a triad relation, which takes the form of the gauge invariance or the gauge variance. When the ternary social networks evolving to the sensitive state, any small disturbance may result in an avalanche. Since the number of triad relations changed in the avalanche being no scale preferred, it should satisfy the power-law distributions both spatially and tempo- rally. Third, for the small-world network, we find out the small-world effect on the occurrence of the avalanche both theoretically and numerically. p$_{ij}$ = ℓ − δ i,j ∑ m<n ℓ − δ m,n . (23) Here δ > 0, and ℓ$_{i,j}$ stands for the smallest distance between i and j on the regular network. Then it at- taches to one neighbor of the selected node at random. Multi-connection and self-connection are forbidden. In this manner, more triangles are created. For the small- world network, we keep q = 0 . 05 constant, and set δ = 1 . 0 and δ = 4 . 0 for comparison. For the size distribution of the avalanche, the exponent increases from -1.85 to -1.69, as shown by Fig. 11 (a) and Fig. 11 (b). Meanwhile, for the time distribution of the avalanche, the exponent in- creases from -2.56 to -2.22, as shown by Fig. 12 (a) and Fig. 12 (b). In a word, the size distribution of the avalanche always satisfies the power-law form, no matter on the regular network, the small-world network or the random net- work. But for the time distribution of the avalanche, it deviates from the power-law form on the random net- work, which we want to do further study in the later work.