On the Tanner Graph Cycle Distribution of Random LDPC, Random\n Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles
Bibliographic record
Abstract
In this paper, we study the cycle distribution of random low-density\nparity-check (LDPC) codes, randomly constructed protograph-based LDPC codes,\nand random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite\ngraph, with a given (irregular) degree distribution, the distributions of\ncycles of different length tend to independent Poisson distributions, as the\nsize of the graph tends to infinity. We derive asymptotic upper and lower\nbounds on the expected values of the Poisson distributions that are independent\nof the size of the graph, and only depend on the degree distribution and the\ncycle length. For a random lift of a bi-regular protograph, we prove that the\nasymptotic cycle distributions are essentially the same as those of random\nbipartite graphs as long as the degree distributions are identical. For random\nQC-LDPC codes, however, we show that the cycle distribution can be quite\ndifferent from the other two categories. In particular, depending on the\nprotograph and the value of $c$, the expected number of cycles of length $c$,\nin this case, can be either $\\Theta(N)$ or $\\Theta(1)$, where $N$ is the\nlifting degree (code length). We also provide numerical results that match our\ntheoretical derivations. Our results provide a theoretical foundation for\nemperical results that were reported in the literature but were not\nwell-justified. They can also be used for the analysis and design of LDPC codes\nand associated algorithms that are based on cycles.\n
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.005 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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; both teacher heads agree on what is shown here.
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".