There are noi such measurements I guess. Only thing you can say is that PO is available at any time when Dev Team has questions. Next to that they should meet for backlog refinement sessions so there should be always interaction throughout a sprint.
Dev Team is the only way and more likely this will also reflect in your reports which eventually turn out to be bad for such miscommunication . Sprint review will also reflect some of its sign as well.
Real world , Dev team sometimes doesn't raise such issues easily because of chain of command. SM needs to handle this issue really diligently.
What does "interacting" mean in your question? I do not see any problem when product owners are interacting with developers because they might be answering questions or clarifications from the developers along the way.
What factors determine if the product owner is interacting with the Dev team throughout the sprint?
I have a feeling that there is something missing in this question. The way it stands now, simple communication would be evident of interaction. Are you asking whether or not the interactions are enough, too much, efficient, wanted?