In the current Bitcoin scheme, one can't accept transactions until it has been incorporated into a block. Suppose two transactions "spending the same coins" enter the network at different points. On average, half the network will have one transaction and half the other. The only way out of this deadlock is which happens to make it into a block first. So you can see that the race across the network is unimportant but the race to get into a block is the deciding factor.