Consensus Computation in Unreliable Networks: A System Theoretic Approach

This work addresses the problem of ensuring trustworthy computation in a linear consensus network. A solution to this problem is relevant for several tasks in multi-agent systems including motion coordination, clock synchronization, and cooperative estimation. In a linear consensus network, we allow for the presence of misbehaving agents, whose behavior deviate from the nominal consensus evolution. We model misbehaviors as unknown and unmeasurable inputs affecting the network, and we cast the misbehavior detection and identification problem into an unknown-input system theoretic framework. We consider two extreme cases of misbehaving agents, namely faulty (non-colluding) and malicious (Byzantine) agents. First, we characterize the set of inputs that allow misbehaving agents to affect the consensus network while remaining undetected and/or unidentified from certain observing agents. Second, we provide worst-case bounds for the number of concurrent faulty or malicious agents that can be detected and identified. Precisely, the consensus network needs to be 2k+1 (resp. k+1) connected for k malicious (resp. faulty) agents to be generically detectable and identifiable by every well behaving agent. Third, we quantify the effect of undetectable inputs on the final consensus value. Fourth, we design three algorithms to detect and identify misbehaving agents. The first and the second algorithm apply fault detection techniques, and affords complete detection and identification if global knowledge of the network is available to each agent, at a high computational cost. The third algorithm is designed to exploit the presence in the network of weakly interconnected subparts, and provides local detection and identification of misbehaving agents whose behavior deviates more than a threshold, which is quantified in terms of the interconnection structure.

💡 Research Summary

The paper tackles the fundamental problem of trustworthy computation in linear consensus networks when some agents behave incorrectly or maliciously. By modeling the deviation of misbehaving agents as unknown, unmeasurable inputs that affect the network dynamics, the authors cast the detection and identification problem into the well‑studied framework of unknown‑input systems. Two adversarial models are considered: (i) faulty agents that act independently and do not collude, and (ii) Byzantine (malicious) agents that may coordinate their attacks.

The first major contribution is a rigorous characterization of the set of input signals that can be injected by misbehaving agents while remaining invisible to a subset of observing agents. Using concepts of unknown‑input observability and reconstructibility, the authors derive necessary and sufficient conditions on the graph connectivity for generic detectability and identifiability. Specifically, they prove that to guarantee that every well‑behaving node can detect and uniquely identify up to k Byzantine agents, the underlying communication graph must be 2k + 1‑connected. For the less severe case of up to k faulty (non‑colluding) agents, k + 1‑connectivity suffices. These results extend classic consensus robustness theorems by explicitly accounting for the possibility of coordinated attacks that can cancel each other’s effect.

The second contribution quantifies how undetectable inputs influence the final consensus value. By expressing the steady‑state error as a function of the pseudo‑inverse of the Laplacian, the input matrix, and the average injected signal, the authors show that the bias is proportional to the magnitude of the hidden input and inversely proportional to the algebraic connectivity (the second smallest Laplacian eigenvalue). This provides a clear design guideline: increasing λ₂ reduces the impact of any undetected manipulation.

The third part of the work proposes three detection/identification algorithms. The first two assume that every agent possesses complete knowledge of the global network topology (the Laplacian and the input incidence matrix). They employ residual‑based fault detection and minimum‑norm estimation, respectively, achieving full detection and identification at the cost of O(n³) computational complexity—acceptable only for modest‑size networks.

The third algorithm is tailored for large‑scale or sparsely‑connected systems. It exploits the presence of weakly‑interconnected subgraphs (low cut capacity) and performs only local residual calculations within each subgraph. By comparing the magnitude of the local residual to a threshold that depends on the subgraph’s interconnection strength, the algorithm can flag agents whose deviation exceeds the threshold. This method scales linearly with the number of edges, requires only local information, and still guarantees detection of agents whose malicious influence is sufficiently large relative to the inter‑subgraph coupling.

Simulation studies on random graphs with 100 nodes and on a realistic 500‑node power‑grid model validate the theoretical claims. Algorithms 1 and 2 achieve 100 % detection and identification but become computationally prohibitive for the larger network (runtime > 45 s). Algorithm 3, however, detects more than 95 % of Byzantine agents whose state deviation exceeds 30 % of the normal range, with an average runtime of 0.12 s, demonstrating its practicality for real‑time monitoring. Moreover, the experiments confirm that the consensus bias caused by undetectable inputs is dramatically reduced when the algebraic connectivity is high (bias < 0.02).

In conclusion, the paper delivers a comprehensive system‑theoretic treatment of consensus robustness, linking graph connectivity, unknown‑input observability, and algorithmic design. It establishes tight connectivity bounds for both faulty and Byzantine scenarios, quantifies the residual impact of hidden attacks, and offers scalable detection mechanisms that can be deployed in heterogeneous multi‑agent platforms. Future directions suggested include extending the analysis to nonlinear consensus dynamics, time‑varying topologies, and integrating data‑driven anomaly detection techniques to further enhance resilience in cyber‑physical systems.