DAG (e.g. Iota)The conceptual idea is that signed transactions reference a prior transaction (which referenced a prior transaction, etc). Thus transactions try to be on the longest chain of transactions. It is also possible to reference multiple chains thus combining chains into one chain.
But since there is no way to determine which chain is the authority, if there is a double-spend on more than one chain then it is not possible to determine which chain to discard. Since chains are not registered in blocks, there is no way to determine which double-spend was issued first.
The only way to resolve this is to have centralized servers which are trusted to cooperate and organize around a single chain. Thus this is just a centralized system (or if not, then a divergent Sybil forked chaos).
In theory if every signer could be an "always on" full node then propagation could be used to order transactions but this again would require all nodes to see the same results from propagation, which is impossible, since an attacker can propagate such that different nodes see different orderings of arrival. Besides not every signer can be online all the time. Again this can be delegated to trusted servers and thus a DAG devolves to a centralized system of cooperating servers.
I did forget to mention on the DAG post, that PoW can be aggregated along with each transaction so that the chain of the DAG with the longest PoW is the unambiguous consensus authority (probably what you mean by "energy"). Then it is a variant of what
I proposed as an idea of improving on Satoshi's PoW scheme.
Nevertheless I can't see that this variant has any advantages over the one I am contemplating that uses blocks. For example, you I think wrote before that Iota can't expand the money supply thus the supply of coins will shrink to zero as users lose passwords over time (apparently Bitcoin and all Proof-of-Stake coins have this same flaw, but not all PoW coins have this flaw). And this will become a serious problem with microtransaction coins because many users will be frivolous. Intense deflation is very destructive to currency.
Also as I had mentioned to you in the past in one of the other threads, there is no force driving consensus. It is possible for cliques to decide it is in their advantage to race each other rather than provide clarity by referencing each other (say for example each clique controls roughly the same amount of hashrate). Whereas, blocks force that there can be only one global partition. And I don't see what advantage getting rid of the blocks provides? There are ways to achieve instant transactions and scaling with blocks. With blocks the calculation of irreversibility is more concrete than with DAG. DAG can't guarantee it will always converge quickly thus it doesn't really guarantee instant transactions, yet once it converges then it might be able to "confirm" a transaction faster than blocks (but I not sure about that since Bitcoin's block period could be reduced if not so much data needs to be sent on block announcement so that orphan rate won't be a concern).
I don't see the advantage since full nodes in Iota will still need to watch all the global transactions if they wish to converge on a single longest chain. And if not then chaos and double-spends on multiple partitions.
What am I missing?