Explaining why PoW with mining is more secure than other methods boils down to the math being the trusted element. This is also the case with Byteball (now called Obyte). The Obyte whitepaper explains that it considers PoW mining to be a leak of network value; to power companies, etc. The Obyte consensus protocol follows a deterministic set of rules. The role of the witnesses is only to order or sequence the units in a fair FIFO manner. Remember, in a DAG an unbounded number of units can be created at the same time, so in the case of an attempted double-spend, the fair ordering will have been determined by the witnesses broadcasting their encounter with the unit as they occur; and not by the amount of fee paid to prioritise them as is the case in BTC. Witnesses do not have other powers and should not be compared with bitcoin miners; they are trusted to act rationally to protect their own reputation in the real world. Full nodes are anonymous nodes which execute the consensus protocol to validate and store the unit. The consensus protocol they execute is deterministic (while that of Bitcoin is probabilistic) so once a unit is confirmed stable in Obyte, it is final; it is then not just extremely unlikely to be invalidated as is the case with Bitcoin after x-number of confirmations; in Obyte it is 100% impossible. The Obyte network is designed to allow for an unlimited number of witnesses, but twelve per node.
Anyone can set-up a witnessing node with relatively little hardware, etc. The Bittrex witness node at address QR542JXX7VJ5UJOZDKHTJCXAYWOATID2 is used by anyone who trusts it to be honest and reliable about broadcasting units as it sees them. They use it, but maybe others don't. Bittrex has a strong logical incentive to run an honest, reliable witness and Bittrex partners, customers etc. transacting Obyte Bytes with them expect Bittrex to protect their brand. If however, they were to fail at that, it would just mean that other nodes would 'vote' away from them by not using them as a witness. A transaction needs just seven of twelve witnesses to confirm sight of the unit. So, it would require seven simultaneous 'failed' witnesses (for a given full node) for a unit to be stuck in an unstable position. Or seven colluding witness for a given full node, for the units to be sequenced in an unfair order. The probability of that is considered extremely unlikely, especially in a mature network.

It's interesting that so many believe that the original Bitcoin way is the only way to achieve Byzantine Fault Tolerance. It is not. Explaining why PoW with mining is more secure than other methods boils down to the math being the trusted element. This is also the case with Byteball (now called Obyte). The Obyte whitepaper explains that it considers PoW mining to be a leak of network value; to power companies, etc. The Obyte consensus protocol follows a deterministic set of rules. The role of the witnesses is only to order or sequence the units in a fair FIFO manner. Remember, in a DAG an unbounded number of units can be created at the same time, so in the case of an attempted double-spend, the fair ordering will have been determined by the witnesses broadcasting their encounter with the unit as they occur; and not by the amount of fee paid to prioritise them as is the case in BTC. Witnesses do not have other powers and should not be compared with bitcoin miners; they are trusted to act rationally to protect their own reputation in the real world. Full nodes are anonymous nodes which execute the consensus protocol to validate and store the unit. The consensus protocol they execute is deterministic (while that of Bitcoin is probabilistic) so once a unit is confirmed stable in Obyte, it is final; it is then not just extremely unlikely to be invalidated as is the case with Bitcoin after x-number of confirmations; in Obyte it is 100% impossible. The Obyte network is designed to allow for the number of witnesses to be limited only by the number of full nodes on the network, but just twelve are needed per transaction.
Anyone can set-up a witnessing node with relatively little hardware, etc. The Bittrex witness node at address QR542JXX7VJ5UJOZDKHTJCXAYWOATID2 is used by anyone who trusts it to be honest and reliable about broadcasting units as it sees them. They use it, but maybe others don't. Bittrex has a strong logical incentive to run an honest, reliable witness and Bittrex partners, customers etc. transacting Obyte Bytes with them expect Bittrex to protect their brand. If however, they were to fail at that, it would just mean that other nodes would 'vote' away from them by not using them as a witness. A transaction needs just seven of twelve witnesses to confirm sight of the unit. So, it would require seven simultaneous 'failed' witnesses (of a given transaction) for a unit to be stuck in an unstable position, or be sequenced in an unfair order. The probability of that is considered extremely unlikely, especially in a mature network. The white paper explains this in full detail.