Average-case and smoothed analysis of graph isomorphism
Bibliographic record
Abstract
We propose a simple and efficient local algorithm for graph isomorphism which succeeds for a large class of sparse graphs. This algorithm produces a low-depth canonical labeling, which is a labeling of the vertices of the graph that identifies its isomorphism class using vertices’ local neighborhoods. Prior work by Czajka and Pandurangan showed that in the Erdős–Rényi model G(n,pn), the degree profile of a vertex (i.e., the sorted list of the degrees of its neighbors) gives a canonical labeling with high probability when npn=ω(log4(n)/loglogn) (and pn≤1/2); subsequently, Mossel and Ross showed that the same holds when npn=ω(log2(n)). We first show that their analysis essentially cannot be improved: we prove that when npn=o(log2(n)/(loglogn)3), with high probability there exist distinct vertices with isomorphic 2-neighborhoods. Our first main result is a positive counterpart to this, showing that 3-neighborhoods give a canonical labeling when npn≥(1+δ)logn (and pn≤1/2); this improves a recent result of Ding, Ma, Wu and Xu, completing the picture above the connectivity threshold. Our second main result is a smoothed analysis of graph isomorphism, showing that for a large class of deterministic graphs, a small random perturbation of the edge set yields a graph which admits a canonical labeling from 3-neighborhoods, with high probability. While the worst-case complexity of graph isomorphism is still unknown, this shows that graph isomorphism has polynomial smoothed complexity.
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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 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.006 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".