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First we define the node relations in analogy with Ref. [1] with some relaxation. We propose each node in the network has two values, +1 and -1, standing for each person’s two opposite standpoints in the society, or two opposing opinions. For a triad relation, if all nodes have the same value, i.e., all persons have the same opinion, the relation is stable. If there are different opinions in a triad relation, conflicts are easily to occur, so the re- lation is unstable. The stable and unstable triad rela- tions are given in Fig. 1. In our model, the stable and unstable triad relations have the probabilities to change into the other, which is more realistic for the social net- works. As we know in the society, stable relations may become unstable because of conflicts and the surround- ings, and unstable relations may become stable because of the same interest and the surroundings. Considering each node has two values which have the probabilities to change into the other, we study the social networks in association with the spin systems. In the following con- text, we establish our model in association with the Ising model and the generalized Glauber dynamics [17, 18]. 

When associated with the spin systems, each node in the network standing for a spin. We propose the Hamil- tonian of each triangle to be 

<img src='content_image/410900.jpg'>

where σ$_{i}$, σ$_{j}$ and σ$_{k}$ stand for the three spins of the tri- angle. In the binary case, σ$_{i}$, σ$_{j}$ and σ$_{k}$ can take either of the two values, +1 and -1. Besides, α > 0 is a real parameter, standing for the strength of coupling. So the Hamiltonian is the form of the Ising model. The evolving rules are as follows. At every step we choose a triangle from the network randomly, and choose one of the three 

## II. DYNAMIC BALANCE 

## A. The Node Model 

nodes from the selected triangle at random, then let the selected node do spin flipping. The spin flipping proba- bility is defined as 

where H ′ Δ = − α ( σ ′ $_{i}$σ$_{j}$ + σ$_{j}$σ$_{k}$ + σ ′ $_{i}$σ$_{k}$ ) stands for the fi- nal Hamiltonian after the spin flipping, with σ ′ i the final value of the selected spin, and σ$_{j}$ , σ$_{k}$ the values of the other spins unchanged. It is noted that the spin flipping probability depends on the final value, independent of the initial one [18]. In addition, Z = ∑ σ ′ $_{i}$=+1 , − $_{1}$e − βH ′ Δ is the normalizing factor and β is a geometrical temper- ature determined by the structure of the network. Since β is not a real temperature, and in order to embody the majority principle, we propose 

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where N is the size of the network, and M = ∑ i 〈 σ$_{i}$ ( t ) 〉 /N is the average value of all spins. So the spin flipping probability depends both on the triad rela- tion and the surroundings [18, 19]. 

The numbers of positive and negative nodes in the net- work are as follows, 

<img src='content_image/410897.jpg'>

where N is the number of total nodes and a$_{i}$ stands for the triangle density, satisfying ∑ 4 i =1 a$_{i}$ = 1. The densi- ties of each triangle attached to a positive node are 

where i = 1 , 2 , 3, and C$_{i}$ is the number of positive nodes in a$_{i}$ . In a similar way, the densities of each triangle attached to a negative node are 

where j = 2 , 3 , 4, with D$_{j}$ the number of negtive nodes in a$_{j}$ . 

For each node, the spin flipping probabilities from one value to the other are given by Eq. (2), i.e., 

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So the total flipping probabilities for each node from one value to the other are as follows, 

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