I've been wondering how the network would deal with some person with malicious intent who has 20 computers around the world send the same transaction to various people.
Hmm, 20 computers sending the same transaction to various people? I may be misunderstanding you, but are you saying that if all those computers had access to 1 wallet.dat file containing 1 BTC, what would happen if they all tried to send it somewhere?
Such a feat would be nearly impossible, as the attacker has to have mined a block first, then sync all 20 of those computers, prepare the transaction, send it out, all whilst holding the block and risking that someone else may crack a block before he releases it. That all would have to be done in less than 10 minutes.