no expert on PPI networks but there is a high risk of assuming things that are not there (or true).
It is definitely so that P1 can interact with P2 and P2 with P3, but there is no chance (or indication) that P1 can (directly!!!) interact with P3 . (they can for instance bind/target different regions in P2) , transitive reasoning here might be dangerous.
I fit in my pants, my pants fits in my suitcase thus I fit in my suitcase 😉