When can transitivity emerge from cycle-suppressing dynamics?

[DavidMartin]

TL;DR

Can cycle suppression in a directed graph cause transitivity to emerge, yielding a partial order without assuming transitivity from the start?

Let V be a finite set and let R⊆V×V be a directed binary relation.
Assume that R is not assumed to be transitive or acyclic a priori.

Equivalently, think of R as the edge set of a directed graph G=(V,E), possibly containing directed cycles.

I am interested in the following general question:

Suppose we define some structural or dynamical condition on R that penalizes or suppresses directed cycles (for example, by minimizing a functional that increases with the number or length of directed cycles).

Under what conditions can such a process lead to a relation that is:

1. acyclic, and
2. transitive,

thus forming a genuine partial order on V?

In particular:

• Are there known results characterizing when cycle suppression forces a relation to converge (or reduce) to its transitive closure?
• Are there dynamical or variational frameworks on directed graphs where transitivity appears as a stable configuration rather than being imposed axiomatically?
• Is there a known characterization of when acyclicity plus some additional structural constraint implies transitivity?

I am not asking about the transitive closure construction itself (which is well known), but rather about mechanisms or structural conditions under which transitivity can arise from a more general directed relation.

References to work in order theory, graph theory, or discrete dynamical systems would be appreciated.

 

[haruspex]

Suppose we define some structural or dynamical condition on R that penalizes or suppresses directed cycles (for example, by minimizing a functional that increases with the number or length of directed cycles).

You will need to be rather more specific for the question to be interesting.
If you imagine changes to (i.e. deletions from) R in response to some penalty or circuit minimisation pressure then you need to constrain what deletions are permitted. It would be trivial to end up with R empty. Some other pressure needs to act.