I have a bunch of Bitcoin nodes supporting the Bitcoin network. Is there a way to tell if one of my servers is in fact running without installing a 3rd party software. Say I wanted to pay one of my users that hosts a Bitcoin server, but I don't want to pay him unless I can detect his Bitcoin server is indeed running. Is there some type of unique identifier to tell if one of my nodes is running?
Bitcoin nodes do not have a unique identifier, and this is by design.
In fact, any property that allows someone on the network to verify whether two connections (even separated in time) are to the same node, is something we'd call a fingerprinting attack.
As such identification could be used to link transactions coming from the same node together, they are a potential privacy leak.