Considerable attention has been devoted to the evolution of social networks through time. Some has been descriptive, some has sought fundamental principles for studying network evolution and some has been analytical. The work described in this paper represents all three approaches and is cast within the framework of rational choice theory. One foundation is found in the work of Jackson and Wolinsky [JW] [Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks, Journal of Economic Theory, 71, 44-74] who consider actors having some calculus for the costs of maintaining social ties and the benefits received by virtue of being located in a network of ties. By parameterizing the costs and benefits, they derived equilibrium structures under specified combinations of parameters when ties are formed by rational actors. The second foundation is found in the simulations of Hummon [Hummon, N.P., 2000. Utility and Dynamic Social Networks, Social Networks, 22, 221-249.] that were based on the JW work. Hummon found that there were equilibrium structures that were not anticipated in the formal analyses of JW. To resolve this discrepancy, using networks with a fixed set of vertices and using the JW framework, this paper explores the transitions between networks on the lattices of all graphs with a fixed number of vertices through the addition and deletion of ties. While these transitions can be described, a close examination of them reveals the equilibrium structures anticipated by JW, the equilibrium structures located by the Hummon simulations, plus some other equilibria. Modified theorems for the equilibria forms are presented together with a generalization of the equilibrium concept.