The question is: What does happen to the real-world networks which cause them not to grow permanently? The idea here is that real-world networks have to pay the cost of growth. We investigate the growth and trade-off between value and cost in the networks with cost and preferential attachment together. Since the preferential attachment in the BA model does not consider any stop against the infinite growth of networks, we introduce a modified version of preferential attachment of the BA model. This idea makes sense because the growth of real networks may be finite. In the present study, by combining preferential attachment in the science of temporal networks (interval graphs), and, the first-order differential equations of value and cost of making links, the future equilibrium of an evolving network is illustrated. During the process of achieving a winning position, the variables against growth such as the competition cost, besides the internally structural cost may emerge. In the end, by applying this modified model, we found the circumstances in which a trade-off between value and cost emerges.