Discrete-to-Continuum Limits of Graph-Regularized Energy Functionals on Irregular Domains

This paper investigates the discrete-to-continuum convergence of graph-regularized energy functionals defined on sequences of irregular graphs. Such functionals arise in machine learning, data science, and numerical analysis, where graphs serve as discretizations of continuous domains. While Γ-convergence results are well-established for regular graph sequences (e.g., uniform lattices or quasi-uniform point clouds), the behavior under structural irregularity—such as non-uniform vertex distributions, heterogeneous edge weights, and variable local connectivity—remains poorly understood. We introduce a set of mild geometric assumptions that accommodate substantial irregularity while still guaranteeing compactness and variational convergence. Under appropriate scaling of weights and regularization parameters, we prove that the sequence of discrete energies Γ-converges to a continuum limit involving a p-Dirichlet energy and an Lq fidelity term. The analysis reveals a critical scaling window in which the discrete gradient structure approximates the continuum Sobolev norm. Counterexamples demonstrate the sharpness of the assumptions, highlighting how specific irregularities can lead to degeneracy or loss of compactness. Our results provide a rigorous foundation for graph-based variational methods on realistic, irregular domains.