On the complexity of greedy routing in ring-based peer-to-peer networks
Bibliographic record
Abstract
We investigate the complexity of greedy routing in uniform ring-based random graphs, a general model that captures many topologies that have been proposed for peer-to-peer and social networks. In this model the nodes form a ring; for each node u we independently draw the set of distances along the ring from u to its "long-range contacts" from a fixed distribution P (the same for all and connect u to the corresponding nodes as well as its ring successor. We prove that, for any distribution P, in a graph with n nodes and an expected number of long-range contacts per node constructed in this fashion, the expected number of steps for greedy routing is Ω((log2n)/lalog*n), for some constant a > 1. This improves an earlier lower bound of Ω((log2n)/llog log n) by Aspnes et al. and is very close to the upper bound of O((log2n)/l) achieved by greedy routing in Kleinberg's (one-dimensional) "small-world" networks, a particular instance of uniform ring-based random graphs.
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 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 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".