Your diagram still doesn't make sense, sorry, it still appears to bridge out most of the bulbs. Is this line going direct to the add on socket perhaps ?. For current to flow there has to be a complete circuit between the mains live and neutral. Your diagram suggests that the live ( NOT +) feeds the bulbs and stops at the socket and the neutral ( NOT -) feeds the bottom of the socket only, so the current can't flow round thro the bulbs and back to the mains neutral.
If the bulbs are the type that screw in, the way to fix them is to unscrew all the bulbs and check their resistance with your meter, and keep going until you find the blown one ( or 2).
They are not very safe and not allowed in the UK, because if you unscrew one bulb and touch the contacts you will get a full mains shock.
Ken