A homological approach to consensus and fault tolerance in decentralized sensor networks

Abstract

Decentralized sensor networks, such as those governing robotic swarms or IoT infrastructure, face fundamental challenges in achieving global consensus from local measurements, particularly in the presence of faulty or malicious Byzantine nodes. Traditional algorithms, often based on gossip protocols or statistical filtering, can struggle with convergence speed and robustness under high fault rates. This paper introduces a novel framework for decentralized consensus and fault tolerance by leveraging tools from homological algebra. We model the sensor network and its communication topology as a time-varying simplicial complex. Local sensor readings are treated as sections of a data sheaf defined over this complex. We demonstrate that the problem of achieving consensus is equivalent to finding a global cocycle that is locally exact. Inconsistencies introduced by faulty nodes manifest as non-trivial elements in the first cohomology group (H¹) of the sheaf. Our proposed algorithm, "Cohomological Consensus," computes these obstructions in a distributed manner, allowing for the identification and isolation of faulty nodes. Simulations on large-scale networks demonstrate that our method achieves up to 35% faster convergence compared to state-of-the-art gossip algorithms and remains robust to Byzantine failure rates exceeding 40%, a significant improvement over existing methods.