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## III. EVALUATION OF A , B , C FOR SUBNET NETWORKS 

In this section we describe the computer algorithm we use to evaluate expressions of A, B, C for the Potts model with 2- and/or 3-site interactions in subnet networks. 

For the Potts model with 2-site interactions, consider the 2 × 2 subnet network in Fig. 6(a) as an example. The Boltzmann weight is 

<img src='content_image/365699.jpg'>

where v = e K − 1, K is the 2-site coupling of the Potts model. 

Terms in the expansion of the products can be represented by graphs. As shown in Fig. 6(a), there are 9 bonds in the subnet. Define two states for each bond, occupied and vacant, then there are total 2 9 = 512 graphs corresponding to the 512 terms in the expansion of (7). For example, Fig. 6(b) is a graph that corresponds to the term ∑ { 4 , 5 , 6 } vδ$_{1}$$_{,}$$_{6}$vδ$_{2}$$_{,}$$_{6}$ = q $^{2}$v$^{2}$δ$_{1}$$_{,}$$_{2}$ contributing to B with a term q $^{2}$v$^{2}$. 

The 512 graphs are divided into five types according to following rules: 

1. Type-1, graphs with isolated spins 1, 2 and 3. The sum of these graphs generates the expression of A . 

2. Type-2, graphs with spins 1 and 2 connected and spin 3 isolated. The sum of these graphs contributes to the expression of Bδ$_{12}$ . For clarity we denote it as B$_{12}$ . 

3. Type-3, graphs with spins 2 and 3 connected and spin 1 isolated. The sum of these graphs contributes to the expression of Bδ$_{23}$ and denoted as B$_{23}$ . 

4. Type-4, graphs with spins 3 and 1 connected and spin 2 isolated. The sum of these graphs contributes to the expression of Bδ$_{31}$ and denoted as B$_{31}$ . 

<img src='content_image/365697.jpg'>

