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The past decade has seen a flourishing of advances in harmonic analysis of graphs. They lie at the crossroads of graph theory and such analytical tools as graph Laplacians, Markov processes and associated boundaries, analysis of path-space, harmonic analysis, dynamics, and tail-invariant measures. Motivated by recent advances for the special case of Bratteli diagrams, our present focus will be on those graph systems $G$ with the property that the sets of vertices $V$ and edges $E$ admit discrete level structures. A choice of discrete levels in turn leads to new and intriguing discrete-time random-walk models. Our main extension (which greatly expands the earlier analysis of Bratteli diagrams) is the case when the levels in the graph system $G$ under consideration are now allowed to be standard measure spaces. Hence, in the measure framework, we must deal with systems of transition probabilities, as opposed to incidence matrices (for the traditional Bratteli diagrams). The paper is divided into two parts, (i) the special case when the levels are countable discrete systems, and (ii) the (non-atomic) measurable category, i.e., when each level is a prescribed measure space with standard Borel structure. The study of the two cases together is motivated in part by recent new results on graph-limits. Our results depend on a new analysis of certain duality systems for operators in Hilbert space; specifically, one dual system of operator for each level. We prove new results in both cases, (i) and (ii); and we further stress both similarities, and differences, between results and techniques involved in the two cases.