Random graphs with specific degree distribution and giant component size
Bibliographic record
Abstract
Random networks are a powerful tool in the analytical modeling of complex networks as they allow us to write approximate mathematical models for diverse properties and behaviors of complex systems. These models are often used to study stochastic processes like percolation, where the giant connected component breaks down as edges are removed, yet they fail to properly account for the size of that component, even in a deterministic setting where all edges exist. Here, we introduce a simple conceptual step to account for such connectivity constraints in existing models. We distinguish between network neighbors based on two types of connections that can lead, or not, to the giant component, which we refer to as critical and subcritical degrees. The giant component is the largest unique component of a network that scales with the network's size under our model. It is analogous to many properties of interest, such as the largest epidemic possible on a contact network or the connectivity of an infrastructure network. Accounting explicitly for this component also allows us to capture important structural features of the network in a system of only one or two equations. When applied to sparse connected networks, we show that our approach compares favorably with the predictions of state-of-the-art models, like message passing, which require a number of equations that are linear in system size. We discuss potential applications of this simple framework for studying infrastructure networks where connectivity constraints are critical to the function of the system.
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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 | 0.001 | 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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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 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".