content_image/433602.jpg content_image/433603.jpg content_image/433604.jpg content_image/433601.jpg content_image/433605.jpg
tral and negative nodes in the network are as follows, 

<img src='content_image/433602.jpg'>

where a$_{1}$ - a$_{10}$ stand for the densities of each triangle. The densities of each triangle attached to a positive node are as follows, 

where i = 1 , 2 , 3 , 5 , 6 , 8, and E$_{i}$ is the number of positive nodes in a$_{i}$ . In a similar way, the densities of each triangle attached to a neutral node are as follows, 

<img src='content_image/433603.jpg'>

<img src='content_image/433604.jpg'>

where j = 5 , 6 , 7 , 8 , 9 , 10, and F$_{j}$ is the number of neutral nodes in a$_{j}$ . The densities of each triangle attached to a negative node are as follows, 

where k = 2 , 3 , 4 , 6 , 7 , 9, with G$_{k}$ being the the number of negtive nodes in a$_{k}$ . 

We study the ternary network in association with the Potts spin systems. Since the values of each spin are more than two, the spin flipping mechanism changes into the spin transition mechanism. For the spin transition mechanism, the spin takes one of the three values, +1, 0 and -1, depending both on the triad relation and the sur- roundings [18, 19]. The Hamiltonian, the spin transition probabilities and the geometrical temperature are illus- trated by Eqs. (1), (2) and (3) respectively. The total probabilities from one value to the others for each node are as follows, 

<img src='content_image/433601.jpg'>

<img src='content_image/433605.jpg'>

But Eq. (16) is correct only for M ̸ = 0. When M → 0, 

the transition probabilities are defined as (the explana- 

