In this paper we introduce the framework of Partial difference Equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with decentralized control schemes. Both leaderless and leader-follower models are considered. PdEs mimic Partial Differential Equations (PDEs) on graphs and can be studied by introducing concepts of functional analysis strongly inspired to the corresponding ones arising in PDEs theory. We generalize different models proposed in the literature by introducing errors in the agent dynamics and analyze agent coordination through the joint use of PdEs and automatic control tools. Moreover, for the simplest control schemes, we show that the resulting PdEs enjoy properties that are similar to those of well-known PDEs like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing formation properties.