There are actually a number of possible threat models to evaluate. But since you asked about overwhelming "the network", I'll look only at attempts to overwhelm the network as a whole and not talk about attempts to overwhelm a few nodes. (You can overwhelm a node with invalid transactions, but that won't hurt the network as a whole.)
First, nodes will only pass verified, valid transactions to other nodes. And a node will only pass one transaction that attempts to claim a given output. And further, a node will not pass a transaction that claims a new output (one that hasn't been stable for a long time) unless it includes a transaction fee.
You have to assume an attacker will use no transaction fees. Otherwise, he'll go broke pretty soon. So in order to get his transactions relayed, he'll have to use old outputs. The problem is, his transactions will get into mined blocks, consuming his old outputs and replacing them with new ones. With new ones, he won't be able to get his transactions relayed for free.
The net effect is that you can't sustain such an attack for long because you can't have that many old transaction outputs that you can claim, and you'll burn them very quickly because you can only relay one transaction per output.
Effectively, after a short initial burst, at best you can send new transactions as fast as they're included in mined blocks, which might be maybe 100 transactions every 10 minutes. That's not much load.