This work proposes a distributed control scheme for the state agreement problem which can guarantee prescribed performance for the system transient. In particular, i) we consider a set of agents that can exchange information according to a static communication graph, ii) we a priori define time-dependent constraints at the edge's space (errors between agents that exchange information) and iii) we design a distributed controller to guarantee that the errors between the neighboring agents do not violate the constraints. Following this technique the contributions are twofold: a) the convergence rate of the system and the communication structure of the agents' network which are strictly connected can be decoupled, and b) the connectivity properties of the initially formed communication graph are rendered invariant by appropriately designing the prescribed performance bounds. It is also shown how the structure and the parameters of the prescribed performance controller can be chosen in case of connected tree graphs and connected graphs with cycles. Simulation results validate the theoretically proven findings while enlightening the merit of the proposed prescribed performance agreement protocol as compared to the linear one.