What factors determine if the product owner is interacting with the Dev team throughout the sprint?

The Development Team is the only one that can and should answer that question. 

Now, if you would like to provide more information about your question and why you ask it, we might be able to give you some better advice. 

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.

We may have a different meaning of "interacting".

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? 

Determine whether the PO has :

Clearly expressing Product Backlog items

Ordering the items in the Product Backlog to best achieve goals and missions

Optimizing the value of the work the Development Team performs

Ensuring that the Product Backlog is visible, transparent, and clear to all, and shows what the Scrum Team will work on next

Ensuring the Development Team understands items in the Product Backlog to the level needed.

Only the Developers can answer that question. Many people think that Sprint Review is a right place to determine that, however this can be misleading. The right time to observe that, is during the Sprint, as you can see if he is available to answer questions, Daily Scrum can be another way to do so. The outcomes of the Scrum team are affected by the Product Owner’s participation and availability during the Sprint..

PO interacting with throughout the sprint, it can be seems as -

1. Early involvement of dev team by PO/Product management in product discovery, epic/sprint refinement

2. Healthy product backlog, with shovel ready backlog items for next 2/3 sprint

3. Scrum team is familiar with all the items they are considering for sprint planning . Ideally Scrum team does not see Product backlog items first time during sprint planning. This is great sigh for team/scrum health.

4. PO and scrum team is ready and welcoming for all the suggestions and looking forward eagerly to work with each other towards achieving sprint and product goals.

5. Healthy conflict between PO and dev team for betterment of product/product backlog items. A healthy discussion is essential for great team.

6. Team does not wait for a specific event to raise any impediment need to be raised to PO

7. PO understands and works along with dev team and SM in working agreement with view to achieve long term goal in future, instead of working towards short term goals.

8.In case there is sudden PB item requested by PO in the middle of sprint, then PO must understand the purpose of this item promotion by Stakeholder, urgency of PB item, PO can make understand SM and Scrum team about the PB item, and out of way Scrum team can be ready to consider this PB item in case of justified case and in limits of working agreement however Team can drop 1 or 2 items from sprint backlog after discussion with PO to consider requested new PB item after proper refinement and justified urgency.  This case need to be handled without animosity , with all team, this shows a healthy relationship between Dev team and PO.