We address the problem of distributed state estimation of a linear dynamical process in an attack-prone environment. Recent attempts to solve this problem impose stringent redundancy requirements on the measurement and communication resources of the network. In this paper, we take a step towards alleviating such strict requirements by exploring two complementary directions: (i) making a small subset of the nodes immune to attacks, or "trusted", and (ii) incorporating diversity into the network. We define graph-theoretic constructs that formally capture the notions of redundancy, diversity, and trust. Based on these constructs, we develop a resilient estimation algorithm and demonstrate that even relatively sparse networks that either exhibit node-diversity, or contain a small subset of trusted nodes, can be just as resilient to adversarial attacks as more dense networks. Finally, given a finite budget for network design, we focus on characterizing the complexity of (i) selecting a set of trusted nodes, and (ii) allocating diversity, so as to achieve a desired level of robustness. We establish that, unfortunately, each of these problems is NP-complete.

中文翻译：

---

冗余、多样性和信任对弹性分布式状态估计的影响

我们解决了在易受攻击的环境中线性动态过程的分布式状态估计问题。最近解决这个问题的尝试对网络的测量和通信资源提出了严格的冗余要求。在本文中，我们通过探索两个互补的方向来减轻这种严格的要求：（i）使一小部分节点不受攻击或“受信任”，以及（ii）将多样性纳入网络。我们定义了正式捕捉冗余、多样性和信任概念的图论结构。基于这些构造，我们开发了一种弹性估计算法，并证明即使是相对稀疏的网络，要么表现出节点多样性，要么包含一小部分可信节点，可以像更密集的网络一样抵御对抗性攻击。最后，给定网络设计的有限预算，我们专注于表征 (i) 选择一组可信节点和 (ii) 分配多样性的复杂性，以实现所需的稳健性水平。我们确定，不幸的是，这些问题中的每一个都是 NP 完全的。