When a transaction is split into two outputs, how does the network "know" you're the owner of the "change" address? Reading this thread about how transaction outputs are split due to "change" made me wonder how the bitcoin network knows you own that second address. How does your wallet/client display the correct balance, if it's unknown to everyone else that the second address is yours?

I'm assuming it has something to do with the wallet and accounts... if so, how can you find the "account" that corresponds with that address?
The bitcoin network does not know who owns which address. The concept of an address doesn't even exist at the protocol level - it's just an authentication layer on top of the scripting logic in transactions.

And of course your own client does - because it has the private key for change addresses it generated itself. That's all that matters. When spending a transaction output, the client will prove that it holds that private key by creating a digital signature that matches the address. That's all the network cares about.