Efficient Estimation of Triangles in Very Large Graphs
Bibliographic record
Abstract
The number of triangles in a graph is an important metric for understanding the graph. It is also directly related to the clustering coefficient of a graph, which is one of the most important indicator for social networks. Counting the number of triangles is computationally expensive for very large graphs. Hence, estimation is necessary for large graphs, particularly for graphs that are hidden behind searchable interfaces where the graphs in their entirety are not available. For instance, user networks in Twitter and Facebook are not available for third parties to explore their properties directly. This paper proposes a new method to estimate the number of triangles based on random edge sampling. It improves the traditional random edge sampling by probing the edges that have a higher probability of forming triangles. The method outperforms the traditional method consistently, and can be better by orders of magnitude when the graph is very large. The result is demonstrated on 20 graphs, including the largest graphs we can find. More importantly, we proved the improvement ratio, and verified our result on all the datasets. The analytical results are achieved by simplifying the variances of the estimators based on the assumption that the graph is very large. We believe that such big data assumption can lead to interesting results not only in triangle estimation, but also in other sampling problems.
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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".