  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.
