Random election proof of stake There are currently many cryptocurrencies based on PoS (Proof of Stake). All these PoS algorithms differ from each other. Some or maybe all of them (I don't know) implement some kind of random election of nodes who participate in the "bookkeeping-process"/"block-forging-process" via having some amount of stake.

How can a random election work in an trustless decentralized environment?
The only way is to disallow membership changes in the "block-forging-process".  The DPoS used by Bitshares does this for instance, with 101 forgers, as do the PAXOS and Raft algorithms.  Otherwise it is trivially easy to Sybil attack the set of "block-forging-nodes", creating 10,000 indistinguishable, seemingly-valid chains with different forgers.  You can require work to become a member of the forging set, but then you've just reduced the whole thing back to Proof of Work.

You may be interested in my blog on the subject: What's wrong with proof of stake?