Predictability vs. Efficiency of Large-Scale Multi-Agent Systems

Abstract

We want to characterize “(in)efficient” and “(un)predictable” dynamics of large-scale multi-agent systems (MAS). We abstract the collective behaviors of such MAS as appropriate classes of cellular and network automata, and define computational notions of predictability and efficiency of those discrete networks’ dynamics. We say that a discrete network’s dynamics is efficient if it settles quickly into an appropriate stationary pattern, such as a stable (“fixed point”) configuration or a temporal cycle; that is, efficiency for us is synonymous with a short transient chain until some stationary behavior is reached. The (in)efficiency of a deterministic networked dynamical system is related to, but not synonymous with, that network’s (un)predictability. We introduce two computational notions of predictability: local and global. We call network dynamics locally predictable; if given a starting configuration, the convergence to a stationary behavior can be efficiently determined or predicted, without performing a step-by-step simulation—unless such simulation itself can be done efficiently. A network’s dynamics is globally predictable, if we can computationally efficiently characterize all possible dynamics for all initial configurations of the system. By efficiently throughout the paper, we mean in the time at most polynomial in the number of network’s nodes. For discrete networks with deterministic local interactions and finite configuration spaces, all properties of such systems’ dynamics are formally decidable; however, for nontrivial network sizes, particular properties may or may not be possible to determine within reasonable computational resources. We overview computational complexity of the reachability-flavored problems as well the problems of enumeration of different dynamics of a given dynamical system across all possible initial configurations, studied in the context of several classes of cellular automata, Boolean networks, and discrete Hopfield networks. We argue that the reachability problems about those systems’ asymptotic dynamics have to be computationally tractable, in order for a given system to have globally predictable dynamics. We also relate the problem of enumerating all possible dynamical evolutions of a network to its global (un)predictability. We study some interesting examples of cellular automata and Boolean networks whose dynamics can be shown to be (i) both predictable and efficient, (ii) efficient but unpredictable, (iii) predictable yet inefficient, and last but not least, (iv) both unpredictable and inefficient. Lastly, we briefly discuss Boolean networks that are globally unpredictable, yet whose efficiency status is still open.