Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs
Bibliographic record
Abstract
A Coalescing-Branching Random Walk (CoBra) is a natural extension to the standard random walk on a graph. The process starts with one pebble at an arbitrary node. In each round of the process every pebble splits into k pebbles, which are sent to k random neighbors. At the end of the round all pebbles at the same node coalesce into a single pebble. The process is also similar to randomized rumor spreading, with each informed node pushing the rumor to k random neighbors each time it receives a copy of the rumor. Besides its mathematical interest, this process is relevant as an information dissemination primitive and a basic model for the spread of epidemics. We study the cover time of CoBra walks, which is the time until each node has seen at least one pebble. Our main result is a bound of O (φ–1 log n) rounds with high probability on the cover time of a CoBra walk with k = 2 on any regular graph with n nodes and conductance φ. This bound improves upon all previous bounds in terms of graph expansion parameters (Dutta et al. [13], Mitzenmacher et al. [27], Cooper et al. [8, 9]). Moreover, we show that for any connected regular graph the cover time is O (n log n) with high probability, independently of the expansion. Both bounds are asymptotically tight. Since our bounds coincide with the worst-case time bounds for Push rumor spreading on regular graphs until all nodes are informed, this raises the question whether CoBra walks and Push rumor spreading perform similarly in general. We answer this negatively by separating the cover time of CoBra walks and the rumor spreading time of Push by a super-polylogarithmic factor on a family of tree-like regular graphs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.051 |
| Meta-epidemiology (narrow) | 0.004 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.006 | 0.014 |
| Open science | 0.006 | 0.006 |
| Research integrity | 0.004 | 0.007 |
| Insufficient payload (model declined to judge) | 0.011 | 0.003 |
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 source (direct Gemma or distilled Codex), 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".