In this post, I want to articulate a little idea I’ve had for a while about using PROPs to describe diagrammatic languages of interconnection in a quick way. Really, it’s nothing more than the observation that certain wiring diagram operads arise from finitely presented symmetric monoidal categories. I’m not very familiar with the PROP literature, so I wouldn’t be surprised if this is already known and articulated somewhere, or is at least folklore. If you’ve seen it before, I’d love to see!
Let me say right off the bat that when I use the terms “PROP” and “operad” here, I will always mean their colored (i.e. many object) variants.
The idea is simple. Every PROP P determines a symmetric monoidal category FP, and every symmetric monoidal category M gives rise to an operad OM. Putting this together, we get a functor
\mathsf{PROP} \to \mathsf{SMCat} \to \mathsf{Operad}
That, explicitly, sends a prop P to the operad OFP whose objects are lists of objects in P, and with morphisms OFP(\vec{p}_1, \ldots, \vec{p}_n; \vec{q}) = P(\vec{p}_1, \ldots, \vec{p}_n; \vec{q}) . This formula is a little abusive: on the left we have an op of arity n, where on the right we have a prop of arity \left( \sum_i \mathsf{len}(\vec{p}_i);\, \mathsf{len}(\vec{q})\right). In other words, we have flattened the list of lists into a single list, and mapped out of that.
The point I want to make here is that PROPs of very simple finite presentation give rise to operads of nesting wiring diagrams that can be more difficult to describe in other ways. For example:

If P is the PROP on a single object generated by a special commutative Frobenius algebra, then by this paper FP is the symmetric monoidal category of cospans on finite sets with disjoint union. Therefore, OFP is the operad of (cospan) bubble diagrams, composing by nesting.

If P is the PROP of a “dualizable object”: that is, two objects i and o, two maps w : () \to (i, o) and c : (i, o) \to () satisfying the zigzag laws familiar from an adjunction. Then OFP is the operad of wiring diagrams for traced symmetric monoidal categories: an object is a formal sum ni + mo which represents a box with n input ports and m output ports, so an op will involve putting k boxes inside an outer box, and we are allowed to connect any inner input to any inner output (using c), any inner input to any outer input and same for outputs (using the identities on i and o), and we can wire an outer input straight to an outer output (using w). The zigzag propagate connections through passing wires:
 If P is the PROP on a “heteromorphism between a cocommutative comonoid and a commutative monoid”, then OFP is the operad \mathsf{Lens}(\mathsf{Arity}) of lens based wiring diagrams. I made up the term “heteromorphism” here so let me explain. We have two objects i and o where i is a commutative monoid and o is a commutative comonoid, and a “heteromorphism” c : (i, o) \to () which commutes with the (co)monoid structure in a way best described by string diagrams:
To see how this works, imagine “cutting open each box” in a wiring diagram and stretching it so that all the inner boxes are splayed out on the left and the outer box is splayed on the right:
This won’t work for all theories of interconnection, of course. It only works for those that don’t treat ports on the same box different from ports on distinct boxes, since of course the definition is not sensitive to the boundaries of boxes. This rules out the diagrams for SMCs for example, which allow connection between different boxes but not between the same box. It also rules out special shapes for boxes, like the diagrams for operads where the boxes are corollas and the diagrams form trees.