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where a$_{1}$ to a$_{4}$ stand for each triangle density, and C$_{i}$ and D$_{i}$ the number of positive and negative nodes in a$_{i}$ respectively. When the network gets dynamically bal- anced, the densities of each triad relation are as follows, 

We note the network reaches an explicit state, indepen- dent of the parameter α . And in the dynamically bal- anced state n + = n $^{−}$, where n + and n − stand for the densities of positive and negative nodes respectively. 

where N is the size of the network, and C ( α ) > 0, being a function of α . The numerical results are given in Fig. 2. More details are given in Appendix A 1. 

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The time needed for dynamic balance is 

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There being only two opposing opinions in the social networks is extreme. We propose the neutral opinion should exist, which means the person holding this opinion is indifferent to other opinions. So we propose each node should have three values, +1, 0 and -1, standing for the three opinions. If there are the same opinion in the triad relation except the neutral opinion, the relation is stable. If there are different opinions in the triad relation except the neutral opinion, the relation is unstable. Otherwise the relation is metastable. The simplest way is by judging the Hamiltonian (see Eq. (1)). If H$_{Δ}$ < 0, it is stable. If H$_{Δ}$ > 0, it is unstable. If H$_{Δ}$ = 0, it is metastable. There are four stable relations, three unstable relations, and three metastable relations in the network, as shown by Fig. 1 and Fig. 3. 

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## B. The Generalized Node Model 

Compared with Eq. (4), the numbers of positive, neu- 

