Motivated by the need for robust calculations and distributed quickly in highly dynamic peer-to-peer (P2P) networks, we present the first known and distributed algorithms for the fundamental problem of the agreement distributed in dynamic networks that display strong sparks (i.e., nodes connect and leave the network continuously over time). Our algorithms guarantee a stable consistency, almost everywhere agreed, with high probabilities even in case of high contradictory emigration and work over time which is polylogarithmic in n (this is the stable size of the network). Our first algorithm can tolerate migration up to a pro-temporal stage, sends only a polylogarithmic number of bits per node per no time, and works under an opponent who does not know the random choices of the algorithm. Our second algorithm, designed for the most demanding adaptive opponent, can tolerate a leak of 1,000,000. Because our algorithms are easy to implement, they could be building blocks for other non-trivial calculations in dynamic networks. A fundamental problem with distributed and multi-agent computing systems is to achieve the overall reliability of the system in the presence of a number of defective processes. This often requires coordination of processes to reach consensus or to agree on a data value needed during the calculation. Examples of consensual applications include agreement on which transactions to transfer to a database, on the order and order in which the computer must be copied and sent to the atom. Practical applications that often require consensus are cloud computing, measurement synchronization, PageRank, opinion formation, smart grids, state estimation, drone control (and multiple robots/agents in general), load compensation, blockchain and others. Bitcoin uses proof of work to achieve consensus without permission in its open peer-to-peer network. To extend the bitcoin blockchain or distributed ledgers, miners try to solve a cryptographic puzzle where the probability of finding a solution is proportional to the computational effort spent on hash per second. The node that first solves such an enigma has its proposed version of the next block of transactions, which will be added to the ledger and eventually accepted by all other nodes. As any node in the network can try to solve the problem of proof of work, a sybil attack is not possible unless the attacker has more than 50% of the network`s computing resources.

In a fully asynchronous system, in which at least one process can have a fall error, the famous result of FLP impossibility has shown that a deterministic algorithm is impossible to obtain by consensus. [5] This impossibility results from the most pessimistic planning scenarios that, in practice, are unlikely, except in conflicting situations such as an intelligent denial of service attacker on the network. In most normal situations, process planning has some degree of natural coincidence. [4] However, we show that a gracefully humiliating consensus, which, under unfavorable network conditions, degrades into a general k-set agreement, allows to face stronger opponents of information: we offer a k-universal algorithm k-set of agreement, Where the number of decision values at the k system scale is not coded in the algorithm, but is determined by the actual power of the opponent in a race: Our algorithm guarantees maximum decision values in k under a VSSC message opponent (n,d) that combines VSSC (n,d) (with a small value of d that ensures cessation) with a certain flow of information. Since the resulting impossibility results show that an opponent of the VSSC message (k,d) is too strong to resolve k-set chords and that a certain flow of information between VSSCs is also required for this purpose, our results are an important step towards the exact solution/limit of impossibility of a general k-set agreement in the directed dynamic networks.