There has recently been a surge of interest in the computational and complexity properties of the population model, which assumes $n$ anonymous, computationally-bounded nodes, interacting at random, and attempting to jointly compute global predicates. Significant work has gone towards investigating majority and consensus dynamics in this model: assuming that each node is initially in one of two states $X$ or $Y$, determine which state had higher initial count. In this paper, we consider a natural generalization of majority/consensus, which we call comparison. We are given two baseline states, $X_0$ and $Y_0$, present in any initial configuration in fixed, possibly small counts. Importantly, one of these states has higher count than the other: we will assume $|X_0| \ge C |Y_0|$ for some constant $C$. The challenge is to design a protocol which can quickly and reliably decide on which of the baseline states $X_0$ and $Y_0$ has higher initial count. We propose a simple algorithm solving comparison: the baseline algorithm uses $O(\log n)$ states per node, and converges in $O(\log n)$ (parallel) time, with high probability, to a state where whole population votes on opinions $X$ or $Y$ at rates proportional to initial $|X_0|$ vs. $|Y_0|$ concentrations. We then describe how such output can be then used to solve comparison. The algorithm is self-stabilizing, in the sense that it converges to the correct decision even if the relative counts of baseline states $X_0$ and $Y_0$ change dynamically during the execution, and leak-robust, in the sense that it can withstand spurious faulty reactions. Our analysis relies on a new martingale concentration result which relates the evolution of a population protocol to its expected (steady-state) analysis, which should be broadly applicable in the context of population protocols and opinion dynamics.
翻译:最近,人们对人口模型的计算和复杂特性产生了兴趣,该模型假定的是匿名美元,计算时的节点为0美元,随机互动,并试图共同计算全球上游。在这个模型中,对多数和共识动态进行了大量调查:假设每个节点最初位于两个州之一的X美元或Y美元,确定哪个州初始数较高。在本文中,我们考虑的是多数/共识的自然概括性,我们称之为比较。我们有两个基准州,即X美元和Y美元,在任何初始配置中,可能是小规模的。重要的是,其中一个州比其他州要高点:我们假设每个节点最初位于两个州之一的x美元或Y美元,确定哪个州最初计算值的多数/共识性。我们面临的挑战是设计一个协议,这个协议可以快速可靠地决定哪个基准国的美元为0美元,然后是美元,在最初计算时,我们建议一个简单的算法比较:基准运算法使用美元(x美元) 美元,然后用美元相对数的初始数,然后用美元,在直方的数值分析中,一个U值的概率值的数值,然后算出一个数值。