Link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequences
Bibliographic record
Abstract
The simplest null models for networks, used to distinguish significant features of a particular network from a priori expected features, are random ensembles with the degree sequence fixed by the specific network of interest. These "fixed degree sequence" (FDS) ensembles are, however, famously resistant to analytic attack. In this paper we introduce ensembles with partially-fixed degree sequences (PFDS) and compare analytic results obtained for them with Monte Carlo results for the FDS ensemble. These results include link likelihoods, subgraph likelihoods, and degree correlations. We find that local structural features in the FDS ensemble can be reasonably well estimated by simultaneously fixing only the degrees of a few nodes, in addition to the total number of nodes and links. As test cases we use two protein interaction networks (Escherichia coli, Saccharomyces cerevisiae), the internet on the autonomous system (AS) level, and the World Wide Web. Fixing just the degrees of two nodes gives the mean neighbor degree as a function of node degree, k;{'}_{k} , in agreement with results explicitly obtained from rewiring. For power law degree distributions, we derive the disassortativity analytically. In the PFDS ensemble the partition function can be expanded diagrammatically. We obtain an explicit expression for the link likelihood to lowest order, which reduces in the limit of large, sparse undirected networks with L links and with k_{max}L to the simple formula P(k,k;{'})=kk;{'}(2L+kk;{'}) . In a similar limit, the probability for three nodes to be linked into a triangle reduces to the factorized expression P_{Delta}(k_{1},k_{2},k_{3})=P(k_{1},k_{2})P(k_{1},k_{3})P(k_{2},k_{3}) .
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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".