In statistical mechanics, Boltzmann defined entropy through the number of microscopic configurations consistent with a given set of macroscopic parameters. In probabilistic terms, this corresponds to a large deviation rate function. In these lectures, we will extend this perspective to sparse graphs and matrices, defining and computing an analogue of Boltzmannʼs entropy in this setting. We will demonstrate how this framework provides powerful tools for studying large deviation phenomena, in particular for the spectral properties of sparse random matrices. We will conclude by highlighting (at least) two important open problems in the field.