Social networks have become an ubiquitous part of society, and it is through these connections that most of our information gets disseminated. However, existing tools, such as graph theory, are not well equipped to sufficiently characterize such large-scale, evolving networks. We present a new empirical approach to analyzing this class of networks in which we supplement graph theoretical concepts with traditional dynamical systems techniques. Specifically, we leverage Dynamic Mode Decomposition (DMD) using small cycle counts as the underlying observable data driving the system. This provides a framework for analyzing the dynamics that describes the growth of social networks. We first review traditional graph theory and well-known random-graph models as well as detail our process of generating dynamical systems using cycle counts. We then demonstrate the feasibility of this approach by obtaining a real-world Twitter dataset, generating corresponding synthetic networks using a new retweet random-graph model, and comparing their DMD spectra and reconstruction error.