Distributed Triangle Detection is Hard in Few Rounds
Why this work is in the frame
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Bibliographic record
Abstract
In the distributed triangle detection problem, we have an n-vertex network G = (V, E) with one player for each vertex of the graph who sees the edges incident on the vertex. The players communicate in synchronous rounds using the edges of this network and have a limited bandwidth of O(log n) bits over each edge. The goal is to detect whether or not G contains a triangle as a subgraph in a minimal number of rounds.We prove that any protocol (deterministic or randomized) for distributed triangle detection requires Ω(log log n) rounds of communication. Prior to our work, only one-round lower bounds were known for this problem.The primary technique for proving these types of distributed lower bounds is via reductions from two-party communication complexity. However, it has been known for a while that this approach is provably incapable of establishing any meaningful lower bounds for distributed triangle detection. Our main technical contribution is a new information theoretic argument which combines recent advances on multi-pass graph streaming lower bounds with the point-to-point communication aspects of distributed models, and can be of independent interest.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.005 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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