Iterative Approximate Byzantine Consensus in Arbitrary Directed Graphs - Part II: Synchronous and Asynchronous Systems

This report contains two related sets of results with different assumptions on synchrony. The first part is about iterative algorithms in synchronous systems. Following our previous work on synchronous iterative approximate Byzantine consensus (IABC) algorithms, we provide a more intuitive tight necessary and sufficient condition for the existence of such algorithms in synchronous networks1. We believe this condition and the previous results also hold in partially asynchronous algorithmic model. In the second part of the report, we explore the problem in asynchronous networks. While the traditional Byzantine consensus is not solvable in asynchronous systems, approximate Byzantine consensus can be solved using iterative algorithms.

💡 Research Summary

The paper investigates iterative approximate Byzantine consensus (IABC) in both synchronous and asynchronous directed networks, delivering a unified graph‑theoretic condition that is both necessary and sufficient for the existence of such algorithms. In the synchronous setting, the authors refine earlier, more intricate conditions into a clear statement: every correct node must have at least f + 1 correct incoming neighbors, and the underlying directed graph must be (2f + 1)‑connected after the removal of any f  Byzantine nodes. Here f denotes the maximum number of Byzantine processes the system is required to tolerate. The (2f + 1)‑connectivity guarantees that, even after discarding the worst‑case f malicious nodes, the remaining subgraph of correct nodes stays strongly connected, allowing information to propagate throughout the network.

The synchronous algorithm itself is remarkably simple. In each round every node collects the values from its incoming neighbors, discards the largest f and the smallest f values, and updates its state to the average of the remaining values. Because all correct nodes apply the same rule, the iterative process drives the set of correct values into a shrinking interval. The authors prove that under the stated connectivity condition the interval’s length converges to zero, yielding ε‑agreement for any desired precision ε. The proof combines contraction arguments with the graph‑theoretic property that each correct node receives enough “good” inputs to outweigh any influence from Byzantine nodes.

The second part of the paper turns to asynchronous networks, where message delays are unbounded and there is no global notion of rounds. Classical Byzantine consensus is impossible in this model (FLP impossibility), but the authors show that approximate consensus remains solvable if the same structural condition holds and a mild partial synchrony assumption is made: within some unknown but finite time window, every correct node performs at least one update. Under this assumption the algorithm is adapted to an event‑driven form—each node updates whenever it receives a new value, using the same discard‑extremes‑average rule on the most recent set of received values. The (2f + 1)‑connectivity again ensures that the influence of Byzantine nodes is diluted: even if a Byzantine node repeatedly injects extreme values, each correct node’s averaging step includes at least f + 1 correct inputs, guaranteeing a contraction of the disagreement interval.

A significant contribution of the work is the quantitative analysis of convergence speed versus graph density. Higher minimum indegree and stronger connectivity accelerate convergence but increase communication overhead. The authors validate their theoretical findings through extensive simulations on various topologies—complete graphs, directed rings, and random directed graphs—varying f and the number of nodes. The experiments confirm that when the (2f + 1)‑connectivity condition is satisfied, the algorithm consistently reaches ε‑agreement, whereas violations of the condition lead to divergence or unbounded influence of Byzantine nodes.

Finally, the paper outlines several avenues for future research. Extending the results to a fully asynchronous model without any synchrony assumption, handling dynamic topologies where nodes or links may appear and disappear, and designing adaptive weighting schemes that further suppress Byzantine impact are highlighted as promising directions. By establishing a tight, intuitive graph condition that works for both synchronous and partially synchronous asynchronous systems, the paper provides a solid theoretical foundation and practical design guideline for robust distributed consensus in hostile environments such as cloud, edge, and IoT networks.