Yes, although in directed acyclic graphs the 'cycles' manifest as multiple paths to a node.

cycles can be handled in two ways: if you are happy with approximate solutions, loopy BP can give that (still linear, but may take longer and there's parameter tuning), for exact solutions you can rewrite the graph to "carry" dependencies (latest paper by Frey)

Could you link to that paper? And does it have anything to do with the junction tree algorithm?

Thanks!

the Christopher Bishop chapter on graphical models has a good section on junction trees IIRC