Gracefully Degrading Consensus and $k$-Set Agreement in Directed Dynamic Networks
Bibliographic record
Abstract
We study distributed agreement in synchronous directed dynamic networks, where an omniscient message adversary controls the availability of communication links. We prove that consensus is impossible under a message adversary that guarantees weak connectivity only, and introduce vertex-stable root components (VSRCs) as a means for circumventing this impossibility: A VSRC(k, d) message adversary guarantees that, eventually, there is an interval of $d$ consecutive rounds where every communication graph contains at most $k$ strongly (dynamic) connected components consisting of the same processes, which have at most outgoing links to the remaining processes. We present a consensus algorithm that works correctly under a VSRC(1, 4H + 2) message adversary, where $H$ is the dynamic causal network diameter. On the other hand, we show that consensus is impossible against a VSRC(1, H - 1) or a VSRC(2, $\infty$) message adversary, revealing that there is not much hope to deal with stronger message adversaries. However, we show that gracefully degrading consensus, which degrades to general $k$-set agreement in case of unfavourable network conditions, is feasible against stronger message adversaries: We provide a $k$-uniform $k$-set agreement algorithm, where the number of system-wide decision values $k$ is not encoded in the algorithm, but rather determined by the actual power of the message adversary in a run: Our algorithm guarantees at most $k$ decision values under a VSRC(n, d) + MAJINF(k) message adversary, which combines VSRC(n, d) (for some small $d$, ensuring termination) with some information flow guarantee MAJINF(k) between certain VSRCs (ensuring $k$-agreement). Our results provide a significant step towards the exact solvability/impossibility border of general $k$-set agreement in directed dynamic networks.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".