Intelligent reflecting surfaces (IRSs) are an emerging technology for future wireless communication. The vast majority of recent research on IRS has focused on system level optimizations. However, developing straightforward and tractable electromagnetic (EM) models that are suitable for IRS 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 IRS architectures based on group and fully connected reconfigurable impedance networks, which are more general and more efficient than conventional single connected reconfigurable impedance network. In addition, the scaling law of the received signal power of an IRS 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 IRS 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.