We introduce the framework of Partial difference Equations (PdEs) over graphs for analyzing the behavior of mobile agents formations equipped with decentralized control schemes. PdEs mimic Partial Differential Equations (PDEs) on graphs and can be studied by introducing functional analysis concepts strongly inspired to the corresponding ones arising in PDEs theory. Different formation models will be considered and agent coordination will be analyzed through the joint use of PdEs and automatic control tools. For the simplest control schemes, it will be shown that the resulting PdEs enjoy properties completely analogous to those of well-known PDEs like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing formation properties.
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