Weak disorder asymptotics in the stochastic mean-field model of distance
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Bibliographic record
Abstract
In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is Θ(log n) and satisfies a central limit theorem with asymptotic means and variances of order Θ(log n), with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution Es for some parameter s > 0, where E is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are s log n and s2 log n, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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