Mobile social robots aimed at interacting with and assisting humans in pedestrian areas need to understand the dynamics of pedestrian social interaction. In this work, we investigate the effect of interaction on pedestrian group motion by defining three motion models to represent (1) interpersonal-distance, (2) relative orientation and (3) absolute difference of velocities; and model them using a dataset of 12000+ pedestrian trajectories recorded in uncontrolled settings. Our contributions include: (i) Demonstrating that interaction has a prominent effect on the empirical distributions of the proposed joint motion attributes, where increasing levels of interaction lead to more regular behavior (ii) Developing analytic motion models of such distributions and reflect the effect of interaction on model parameters (iii) Detecting the social groups in a crowd with almost perfect accuracy utilizing the proposed models, despite the constant flow direction in the environment which causes unrelated pedestrians to move in a correlated way, and thus makes group recognition more difficult (iv) Estimating the level of intensity with considerable rates utilizing the proposed models.