Object graph 'integrals'

Is there some mathematical theory that considers ordered 'integrals' over object graphs? I've selectively checked category theory books and papers, and it looks that they are working with graph-to-graph operations, rather than with 'integral' object graph operations. Papers in graph theory considers some path 'integral' operations in ad-hoc manner like in delivery problem, but I've have not found generic considerations for graph-scoped 'integral' operations.

Such graph integral would be high-order function that produce graph function basing on node specific function and it will combine node specific function using some combinator basing on links in graph.

For theory behind dependency injection such graph 'integral' operations are needed, but I've have not found any papers or books on this simple-looking thing, possibly because I'm using wrong keywords. For example, creation of graph of objects and unwinding graph of objects in dependency injection are such 'integral' operations. We need to execute some object-dependent operation for each object in operation-specific order governed by graph. For example, for graph unwind operation we need to destroy objects that have no dependent objects first. For configuration 'refresh' there is a need to update only affected objects.

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Interaction nets

I'm not certain what you mean by 'integral'. However, I think you might be interested in Yves Lafont's work on Interaction Nets.