content_image/379907.jpg content_image/379906.jpg content_image/379908.jpg content_image/379903.jpg content_image/379904.jpg content_image/379905.jpg content_image/379909.jpg content_image/379910.jpg content_image/379911.jpg
We thank H. Zhu for helpful discussions. The work was supported by the National Natural Science Foundation of China (No. 10875012). 

## Appendix A: Distributions of each triad relation and the time needed for dynamic balance 

For the node model, each node has two spin values, +1 and -1. We define one step of time as executing N spin flipping successively at random [1]. Because each spin flipping is linked up with ( N − 1)( N − 2) / 2 variations of the triangles, the variations of the densities of each triad relation with time are as follows, 

<img src='content_image/379907.jpg'>

When the network gets dynamically balanced, we have 

and 

## Acknowledgments 

## 1. The node model 

<img src='content_image/379906.jpg'>

<img src='content_image/379908.jpg'>

From Eqs. (A1), (A2) and (A3), along with the normal- ization ∑ 4 i $_{=1}$a$_{i}$ = 1, we obtain 

We note the distributions of each triad relation indepen- dent of the parameter α , and n + = n − in the dynamically balanced state. 

Defining at any time the density of positive nodes to be ρ , we obtain 

where j is the number of positive nodes in a$_{i}$ and C j 3 is the combinatorial number. We have the new equations as follows, 

Substitute Eqs. (8), (9) and (A5) into Eq. (A6), we obtain 

<img src='content_image/379903.jpg'>

<img src='content_image/379904.jpg'>

<img src='content_image/379905.jpg'>

<img src='content_image/379909.jpg'>

For Eq. (A7), when M ( t ) ∼ 0, after some calculations, we obtain the approximate result, 

where C ( α ) > 0, being a function of α . 

<img src='content_image/379910.jpg'>

For the dynamic balance of the network, we propose 

<img src='content_image/379911.jpg'>

