We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier duality, probability, harmonic functions, sampling/interpolation, and boundary theory. With the use of limit constructions, finite to infinite, and local to global, we outline how our results for infinite graphs may be viewed as extensions of Shannon's theory: Starting with a countable infinite graph $G$, and a suitable fixed positive weight function, we show that there are certain continua (certain ambient sets $X$) extending $G$, and associated notions of interpolation for (Hilbert spaces of) functions on $X$ from their restrictions to the discrete graph $G$.