Important Note: This Answer Is Very Out of Date
So far, the only means we have found to organize a vast number of node clients in an attack-resistant network that produces a virtual computer is to apply cryptographically produced randomness. Of course, Satoshi also relied on randomness by having miners race to solve a current puzzle whose solutions can only be found randomly using brute force computation - then allowing winners to append blocks of Bitcoin transactions to his chain. DFINITY needs stronger and less manipulable randomness that is produced more efficiently in fixed time. Randomness does not only play an important role to ensure that consensus power and rewards are fairly distributed among all miners. Turing-complete blockchains like DFINITY require a higher level of randomness since smart applications may enable high-volume transactions that hinge on aleatory conditions, so that the potential gain of manipulation could be arbitrarily high.
The solution we found is Threshold Relay, which applies cryptography to create randomness on demand of sufficient network participants in a manner that is almost incorruptible, totally unmanipulable and unpredictable. Using Threshold Relay, DFINITY network participants produce a deterministic Verifiable Random Function (or VRF) that powers network organization and processing.