Improved Bounds for Max Consensus in Wireless Networks
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Bibliographic record
Abstract
In consensus problems, the goal is for the nodes of a network to converge to a certain quantity or a function of their values using local communications. In the maximum value consensus problem, the objective of these communications is for all the nodes to converge to the maximum of their initial values. There are two existing algorithms for the maximum value consensus problem in asynchronous networks: RANDOM-PAIRWISE-MAX and RANDOM-BROADCAST-MAX for which the bounds on the mean convergence time have been derived in the literature. In this paper, we derive tighter bounds on the expected convergence time of these two algorithms when run on grid networks and random geometric graphs, respectively-two models commonly used to capture salient properties of wireless networks. We show that RANDOM-PAIRWISE-MAX run on a 2-D grid graph with n nodes converges in expectation after O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> ) iterations, and RANDOM-BROADCAST-MAX run on a random geometric graph with n nodes converges in expectation after O((n/ log n) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> ) iterations. These bounds improve over the previous best-known upper bounds by factors of √n log n and log n + log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n, respectively. Experiments illustrate that the proposed bounds can be up to 95% tighter than the previous state-of-the-art bounds. Furthermore, we enhance the proposed bounds by introducing probabilistic network link failures, e.g., to model packet drops in wireless networks.
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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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it