Strongly coupled Dyson–Schwinger equations generate infinite power series of running coupling constants together with Feynman diagrams with increasing loop orders as coefficients. Theory of graphons for sparse graphs can address a new useful approach for the study of graph limits of sequences of partial sums corresponding to these infinite power series in the context of Feynman graphons and the cut-distance topology. Graphon models enable us to associate some new analytic graphs to non-perturbative solutions of Dyson–Schwinger equations. Homomorphism densities of Feynman graphons provide a new way of analyzing non-perturbative phase transitions. We explain the structures of topological renormalization quotient Hopf algebras of Feynman graphons which encode gauge symmetries Hopf ideals in the context of the weakly isomorphic equivalence classes corresponding to the Slavnov–Taylor / Ward–Takahashi elements. We apply Feynman graphon representations of combinatorial Dyson–Schwinger equations underlying the Connes–Kreimer renormalization Hopf algebra to construct a new class of Banach bundles for the study of the dynamics of non-perturbative phases in strongly coupled gauge field theories.