Equivalent String Networks and Uniqueness of BPS States

We analyze string networks in 7-brane configurations in IIB string theory. We introduce a complex parameter M characterizing equivalence classes of networks on a fixed 7-brane background and specifying the BPS mass of the network as M_{BPS} = | M |. We show that M can be calculated without knowing the particular representative of the BPS state. Based on detailed examination of backgrounds with three and four 7-branes we argue that equivalent networks may not be simultaneously BPS, an essential requirement of consistency.