Dynamic games are considered with large subpopulations distributed over large sparse graphs, where each agent has mean field coupling with all agents within the same cluster and meanwhile receives impact from neighboring clusters. Tractable limit models are obtained via the network limits of graph Laplacian operators modelling second order interactions. The resulting mean field game (MFG) has what is called Laplexion dynamics. For ring and torus topology cases, solutions for linear quadratic Laplexion MFG systems with infinite populations on infinite node networks are obtained via a second order linear partial differential equation system. This work focusses on the relationship between the finite population model and the sparse network limit model. We prove an \(\epsilon\)-Nash equilibrium property for the obtained decentralized strategies.