[ If this question is too free or speculative, feel free to demote it...]

Graphs typically belong to perturbative QFT as Feynman diagrams, while non-perturbative formulations like lattice QCD have arrays of cells and edges between cells as its fundamental structure.

Does it make any sense to formulate a version of a QFT as a non-perturbative, connected graph structure with a suitable ruleset (not directly arising as partial integrals in an expansion in powers of the coupling)? For example, a lattice QCD's generated configurations could each be illustrated as a graph, if you translate the cells and edges suitably.

Note that I'm considering graphs that have SOME cutoff, so they stay comparable to a lattice formulation from a computational point of view, but the idea would be if there could be some kind of advantage of such a graph for analysis, intuition or computation...