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 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$_{31}$ . 6 1 2 3 4 5 6 1 2 3 4 5 6 (b) (a) FIG. 6: (a) 2 × 2 subnet with pure 2-site interactions. (b) Red (bold) bonds are occupied, other bonds are vacant. W$_{△}$ = ∑ { 4 , 5 , 6 } (1 + vδ$_{1}$$_{,}$$_{6}$ )(1 + vδ$_{1}$$_{,}$$_{5}$ )(1 + vδ$_{2}$$_{,}$$_{4}$ $^{)}$· · · , (7)