Ternary Social Networks: Dynamic Balance and Self-Organized Criticality Qing-Kuan Meng, 1, ∗ Wei Liu, 2, † and Jian-Yang Zhu 1, ‡ $^{1}$Department of Physics, Beijing Normal University, Beijing 100875, China $^{2}$School of Science, Xi’an University of Science and Technology, Xi’an 710054, China (Dated: October 30, 2018) Antal et al. [Phys. Rev. E 72 , 036121 (2005)] have studied the balance dynamics on the social networks. In this paper, based on the model proposed by Antal et al. , we improve it and generalize the binary social networks to the ternary social networks. When the social networks get dynamically balanced, we obtain the distributions of each relation and the time needed for dynamic balance. Besides, we study the self-organized criticality on the ternary social networks based on our model. For the ternary social networks evolving to the sensitive state, any small disturbance may result in an avalanche. The occurrence of the avalanche satisfies the power-law form both spatially and temporally. Numerical results verify our theoretical expectations. PACS numbers: 05.65.+b, 87.23.Ge, 89.75.-k I. INTRODUCTION Antal et al. [1] have studied the balance dynamics on the social networks based on the notion of social balance [2, 3]. In the networks, each node is connected to all the others, representing each person knows all the others in the society. Each edge in the networks has two values, +1 and -1. If the edge is +1, it means the two persons are friendly with each other. If the edge is -1, it means the two persons are hostile towards each other. At every step, they choose a triangle randomly from the network. If the product of the three edges is +1, the triangle is stable. Otherwise, if the product is -1, the triangle is un- stable. For the stable triangle, it satisfies (i) the friend of my friend being my friend; (ii) the enemy of my friend being my enemy; (iii) the friend of my enemy being my enemy; and (iv) the enemy of my enemy being my friend. The unstable triangles always try to be stable, but the final state of the network depends on the edge flipping probability p , which is set manually. If p ≥ 1 $_{2}$, the net- work will reach the state of ”paradise”, with all relations being friendly. Several studies around the balance dy- namics have been carried out, including the studies of the university class of triad dynamics [4] and the satisfi- ability problem of computer science [5], etc. In the first half of this paper, at first based on Ref. [1], we map the edge relations to the node relations with some relaxation, so there are two opposing opinions in the network. We suppose for each triad relation in the social networks, it will change from stable to unstable or the re- verse, caused by the change of some persons’ opinions in it. While for the change, the person’s opinion depending both on the other opinions in the triad relation and the opinions all around him. Next, considering in the social This paper is organized as follows. In Sec. II, we map the edge relations to the node relations and generalize the binary networks to the ternary ones. For both the binary and the ternary networks, we obtain the distributions of each triad relation and the time needed for dynamic balance. In Sec. III, we study the SOC on the regular network and the small-world network respectively, and find out the small-world effect on the occurrence of the avalanche. We end this paper with conclusions in Sec. IV. In the latter half of this paper, we study the self- organized criticality (SOC) [6–8] on the ternary social networks. The previous studies of the SOC on complex networks are mainly based on the BTW model [6, 9–14]. In this paper, we establish our model with some differ- ences. As we know, in the society, if one relation changes from one state into anther, it may affect other relations associating with it. The effect may be big or small, de- pending on many internal factors. So in this paper, we simplify this phenomenon, with an eye to the triad bal- ance dynamics, and make two modifications compared to Sec. II. We find under specific conditions the SOC to be observable. Besides, for the small-world network [15, 16], we find the way of construction have an influence on the occurrence of the avalanche. We analyze it theoretically. Numerical results verify our theoretical expectations. networks, there being two opposing opinions is extreme, we introduce the neutral opinion. So the values of each node are generalized to +1, 0 and -1, and the metastable triad relations appear. We find the densities of each triad relation associated with the Hamiltonian and the geomet- rical temperature which is determined by the structure of the network without manual setting. Based on our model, we obtain the distributions of each triad relation and the time needed for dynamic balance both for the binary and ternary cases. $^{∗}$Electronic address: qkmeng@mail.bnu.edu.cn $^{†}$Electronic address: wliuphys@gmail.com $^{‡}$Author to whom correspondence should be addressed; Electronic address: zhujy@bnu.edu.cn arXiv:1001.0309v1 [physics.soc-ph] 2 Jan 2010