Reconfigurable intelligent surfaces (RISs) are an emerging technology for future wireless communication. The vast majority of recent research on RIS has focused on system level optimizations. However, developing straightforward and tractable electromagnetic models that are suitable for RIS aided communication modeling remains an open issue. In this paper, we address this issue and derive communication models by using rigorous scattering parameter network analysis. We also propose new RIS architectures based on group and fully connected reconfigurable impedance networks that can adjust not only the phases but also the magnitudes of the impinging waves, which are more general and more efficient than conventional single connected reconfigurable impedance network that only adjusts the phases of the impinging waves. In addition, the scaling law of the received signal power of an RIS aided system with reconfigurable impedance networks is also derived. Compared with the single connected reconfigurable impedance network, our group and fully connected reconfigurable impedance network can increase the received signal power by up to 62%, or maintain the same received signal power with a number of RIS elements reduced by up to 21%. We also investigate the proposed architecture in deployments with distance-dependent pathloss and Rician fading channel, and show that the proposed group and fully connected reconfigurable impedance networks outperform the single connected case by up to 34% and 48%, respectively.