Studies of real-world complex systems, such as chemical, biological, and epidemiological systems, often involve dynamical systems on network structures. The diversity of network types and variable characteristics complicates the analysis of how patterns emerge in such network dynamics and poses challenges for effective modeling. Motivated by the need to understand the role of graph structures in these problems, we adopt both analytical and empirical perspectives to develop techniques that capture network structures in these dynamical systems problems and further enable us to understand their effects. In this talk, we will present investigations into the influence of networks on dynamics from these two perspectives. First, we will discuss studies of localized stationary patterns in bistable reaction-diffusion systems, focusing on the case of ring lattices and demonstrating how interaction length affects the connectivity of solution branches and how these systems can be studied analytically. Next, we will highlight a social system application to illustrate effective modeling approaches that integrate networks and create opportunities to connect with real-world data.