Understanding and characterizing the emergent behavior of systems with numerous interacting components is typically difficult. This is especially the case when these interactions occur on an inhomogeneous graph, a situation relevant to many systems in bio- and statistical physics. Here we showcase a data driven approach, aimed at optimally compressing the system's information based on an information-theoretic principle. We develop an efficient numerical algorithm applicable to systems on arbitrary static graphs which employs variational estimators of mutual information to find optimal compression. We demonstrate that the optimal compression maps interpretably extract physically relevant local degrees of freedom. This enables us to construct an effective theory of a strongly correlated system on a quasicrystal.