By incorporating some predictive mechanism into a few pinning nodes, we show that convergence procedure to consensus can be substantially accelerated in networks of interconnected dynamic agents while physically maintaining the network topology. Such an acceleration stems from the compression mechanism of the eigenspectrum of the state matrix conferred by the predictive mechanism. This study provides a technical basis for the roles of some predictive mechanisms in widely-spread biological swarms, flocks, and consensus networks. From the engineering application point of view, inclusion of an efficient predictive mechanism allows for a significant increase in the convergence speed towards consensus.