  In this paper, we study finite-agent linear-quadratic games on graphs.
Specifically, we propose a comprehensive framework that extends the existing
literature by incorporating heterogeneous and interpretable player
interactions. Compared to previous works, our model offers a more realistic
depiction of strategic decision-making processes. For general graphs, we
establish the convergence of fictitious play, a widely-used iterative solution
method for determining the Nash equilibrium of our proposed game model.
Notably, under appropriate conditions, this convergence holds true irrespective
of the number of players involved. For vertex-transitive graphs, we develop a
semi-explicit characterization of the Nash equilibrium. Through rigorous
analysis, we demonstrate the well-posedness of this characterization under
certain conditions. We present numerical experiments that validate our
theoretical results and provide insights into the intricate relationship
between various game dynamics and the underlying graph structure.
