What technology could break this relationship so that the cost of securing the network decreases as the value of the network increases?
I'm not sure you need to completely break this relationship, just make it less linear. If I want to go from securing something against amateurs to securing it against governments I might go from 128 to 1024 bit encryption - at roughly 8x the cost to secure. But if an adversary has to use some form of brute force to crack it, they have to do 2^(1024 - 128) or about 10^70 times the work, so it's a lot more than 8x the security.
Obviously the above doesn't apply to Proof of Work, which seemingly require the relationship to be linear. Maybe Proof of Stake might get us in the right direction.