Types of stable network structures based on exhaustive search

Based on the works of Doreian [Doreian, P., 2006. Actor network utilities and network evolution. Social Networks 28, 137–164] and Xie and Cui [Xie, F., Cui, W., 2008b. A note on the paper ‘cost range and the stable network structures’. Social Networks 30, 102–103], the paper applies exhaustive search and inductive approaches to analyze possible transitions among isomeric social networks with five vertices. A total of seven types of stable network structures are identified, which include the complete, star, chain, cycle, near-chain, near-cycle and null graphs. Then, a modified theorem for the equilibrium forms which corrects the previous work is presented.

► Exhaustive search and inductive approach are applied into social networks.
► Modified theorem for the equilibrium social network with 5 vertices is presented.
► Seven types of stable network structures are identified.