We address the problem of reaching resilient consensus among a set of agents in the presence of agents which are either under attack or behaving noisy (called faulty agents). Here, an attacked node is a node that is trying to make the consensus of the network to be driven to a value desired by the attacker, and a noisy node is a node that can be modeled as a random variable. We propose general algorithms to reach consensus in the presence of attacked nodes or noisy nodes. The algorithms are general since they receive as input a consensus algorithm, the network of nodes and initial nodes' values, and the number of maximum allowed attacked or noisy nodes. These algorithms let the agents identify the set of attacked nodes and correct the consensus value by ignoring the faulty nodes' values. We prove that if the number of faulty nodes is below the maximum allowed, then each non-faulty agent detects them with the guarantee of no false positives. If the input consensus algorithm is discrete-time and has polynomial-time complexity $\mathcal{O}\left(C\right)$, then the proposed correction algorithms have polynomial-time complexity $\mathcal{O}\left(C{n}^f\right)$ (and $\mathcal{O}\left(Cn\right)$ for the detection of existing faulty nodes), where the number of nodes of the network is n, and the maximum allowed faulty nodes is f. Finally, we show the effectiveness of the algorithms through simulation, pointing out some attacking scenarios dealt with the aforementioned methods, which could not be tackled by the state-of-the-art. Also, we present examples where two different consensus algorithms are used as input.

我们解决了在存在受到攻击或行为嘈杂的代理（称为故障代理）的情况下在一组代理之间达成弹性共识的问题。在此，被攻击节点是试图使网络的共识被驱动为攻击者所需值的节点，而有噪声节点是可以建模为随机变量的节点。我们提出了通用算法，以在受到攻击的节点或嘈杂的节点存在时达成共识。该算法是通用的，因为它们接收到共识算法，节点网络和初始节点的值以及所允许的最大受攻击或噪声节点数量作为输入。这些算法使代理能够识别受攻击节点的集合，并通过忽略故障节点的值来更正共识值。我们证明，如果故障节点的数量低于允许的最大数量，则每个无故障的代理都会检测到它们，并保证没有误报。如果输入共识算法是离散时间并且具有多项式时间复杂度$\mathcal{Ø}（C）$，则所提出的校正算法具有多项式时间复杂度 $\mathcal{Ø}（C{ñ}^F）$ （和 $\mathcal{Ø}（Cñ）$（用于检测现有的故障节点），其中网络的节点数为n，允许的最大故障节点数为f。最后，我们通过仿真展示了算法的有效性，指出了使用上述方法处理的一些攻击场景，而这些都是最新技术无法解决的。另外，我们提供了使用两个不同的共识算法作为输入的示例。