In the last twenty years, the modeling of complex systems has become a relevant domain of study, not only in applied sciences, but also among mathematicians. The recent improvements in the understanding of interacting particle systems, as well as the new insights coming from graph theory, allow to mathematically tackle new exciting problems in the challenging world of complex phenomena. This thesis addresses a rather general class of interacting particle systems defined on graph sequences. Notably, it focuses on weakly interacting particles described by differential equations, both deterministic and stochastic, where an extra structure encoding the connections among the particles is present. The mean-field hypothesis under which each particle is connected to all the others and in exactly the same way, is relaxed to a much more general assumption: the connections between the particles are supposed to be encoded by a general network, instead of the trivial complete graph of the mean-field case, meaning that a particle is interacting with another in a way that is proportional to the weight of the edge connecting the twos in the underlying graph. Several aspects for this class of models appear to be new in current research and demand new tools and techniques, but also new insights to unveil how the complexity behind the underlying network affects the particle dynamics. The present manuscript poses the focus on three main aspects: the relationship with the mean-field behavior, i.e., on which graph sequences the system behavior is suitably described by the classical mean-field limit; the extensions to inhomogeneous graph sequences and consequent inhomogeneous behaviors; finally, a first study on the long time dynamics for a particular model of interacting diffusions on graphs.