The combination of machine learning (ML) and sparsity-promoting techniques is enabling direct extraction of governing equations from data, revolutionizing computational modeling in diverse fields of science and engineering. The discovered dynamical models could be used to address challenges in climate science, neuroscience, ecology, finance, epidemiology, and beyond. However, most existing sparse identification methods for discovering dynamical systems treat the whole system as one without considering the interactions between subsystems. As a result, such models are not able to capture small changes in the emergent system behavior. To address this issue, we developed a new method called Sparse Identification of Nonlinear Dynamical Systems from Graph-structured data (SINDyG), which incorporates the network structure into sparse regression to identify model parameters that explain the underlying network dynamics. We tested our proposed method using several case studies of neuronal dynamics, where we modeled the macroscopic oscillation of a population of neurons using the extended Stuart-Landau (SL) equation and utilize the SINDyG method to identify the underlying nonlinear dynamics. Our extensive computational experiments validate the improved accuracy and simplicity of discovered network dynamics when compared to the original SINDy approach. The proposed graph-informed penalty can be easily integrated with other symbolic regression algorithms, enhancing model interpretability and performance by incorporating network structure into the regression process.