| Study the following complex adaptive system:(1) Because of the interactive betweenagents in the system, several Local-Worlds, which is small enough from macroscopic viewsuch that there are enough Local-Worlds in the system and is large enough from microscopicview such that there are enough agents inter-acted with each other in the Local-World, areemerged.(2) In a relative short time-scale, the interactive of cooperative stochastic differentialgames in a certain Local-World between agents are defined, however, the interactive ofnon-cooperative stochastic differential games between agents standing on differentialLocal-Worlds. Furthermore, the state, property and character are determined by the system’stopology configuration. In other words, the cooperative stochastic differential between agentsin a certain Local-World make agents behavior synchronize, and the non-cooperativestochastic differential game between agents standing on differential Local-Worlds makeagents behavior not synchronize.(3) In a long time-scale, under preferential attachment,growth and decline mechanism with agents’ intellective, autonomous and society properties,agents in the system can adjust adaptively its owner topological configuration such that thecomplex adaptive system innovation.The objectivity is to analyze the optimal strategies trajectory distribution law the agentsin the system on arbitrary time, and the duration distribution law of the optimal strategiestrajectory, and to master the evolution law of the system such that one can control the systemoperate healthy.The processes&methods are: In time-scale, as for the cooperative stochastic differentialgame between agents in a certain Local-World, firstly, the Local-World can be considered as aunite and its optimization problem should be resolved, then an stochastic dynamical Shapleyimputation mechanism adjusted is introduced to make the agents’ payment in the Local-Worldrational, which make the their behaviors synchronized. As for the non-cooperative stochasticdifferential game between agents standing on different Local-Worlds, it can be looked as theinteractive between different Local-Worlds that can be defined as Super-Agent, then thefeedback Nash equilibrium solution can be analyzed. To make the different solutionsmentioned above identify, a non-linear operator is constructed to couple them to the optimalstrategies trajectory for arbitrary agents in the system, coupled with their optimal payment. Atlast, the optimal strategies trajectory for arbitrary agents in system is proven to beconvergence in the short time-scale, and the corresponding attractor is determined. In the longtime-scale, a coevolution stochastic process that consists of the topological configuration of agents’ interactive, agents’ behaviors, and environment is constructed. Furthermore, there are6sub-processes in this coevolution process, which are updating agent behavior, creating anew interactive with the agents in same Local-World, creating a new interactive with theagents in other same Local-World, deleting an old interactive, creating an interactive with anew agent of system and deleting the agent, which is driven by preferential attachmentmechanism and volatility mechanism with payment of agent. In some sense, the coevolutionsystem can be emerged by these6sub-processes, then, the distribution character of theattractor of the optimal strategy can be obtained by analyzing this stochastic process. In theend, percolation criticality of the complex adaptive is considered, by analyzing theconnectivity of the system by deleting randomly some certain agents and deletingintentionally some certain agents according to the order from large to small of agents’transitory payment respectively, then the critical probability and the system’s criticality can beobtained, which can make the system develop stable.After analyzing, the result is: in the short time-scale, the optimal strategy of arbitraryagents can be determined by considering the character of the cooperative/non-cooperativestochastic differential game, in the long time-scale, agent decide its owner behavior accordingto the invariable distribution of the optimal strategies trajectory that driven by preferentialattachment and volatility mechanism with its payment, which makes this complex adaptivesystem evolve. Furthermore, the system is robust when it is attacked randomly, however, it isvulnerable when it is attacked intentionally.It is conclude that:(1) in short time-scale, the optimal strategy trajectory with non-linearoperator of cooperative/non-cooperative stochastic differential game between agents canmake agents in a certain Local-World coordinate and make the Local-World paymentmaximize, and can make the all Local-Worlds equilibrated; furthermore, the optimal strategyof coupled game can converge into a certain attractor that decide the optimal property in thistime-scale;(2) in long time-scale, the complex adaptive is a coevolution system of agents’behaviors and system’s topology configuration, it is a Markov process, and the correspondinginvariant distribution can be determined by agents’ behaviors, system’s topology configuration,the noise of agent’s behavior and the system population, when the noise tend to0the invariantdistribution can converge into a certain rate function with large deviations principle, and thestate of all possible strategies is a (β, p)correlated equilibrium; when the population ten toinfinity, the invariant distribution can converge into a certain interval with rate function-r~β(σ, q);(3) when the system is attacked randomly, there are at least two large components keep the system connected in this system; however, when the system is attacked intentionally,the system has the critical point with deleting agents, which means that there are two largecomponents keep the system connected in this system if the deleting probability is smallerthan the critical probability and the system cannot be connected if the deleting probability islarger than the critical probability; furthermore, the critical probability is determined by theagents payment. |
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