Cheeger’s Inequalities for Vertex Expansion and Reweighted Eigenvalues
Bibliographic record
Abstract
Abstract. The classic Cheeger’s inequality relates the edge conductance [Formula: see text] of a graph and the second smallest eigenvalue [Formula: see text] of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality [Formula: see text] connecting the vertex expansion [Formula: see text] of a graph [Formula: see text] and the maximum reweighted second smallest eigenvalue [Formula: see text] of the Laplacian matrix. In this work, we first improve their result to [Formula: see text], where [Formula: see text] is the maximum degree in [Formula: see text], which is optimal up to a constant factor. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analogue in relating vertex expansions and reweighted eigenvalues. These include the following: (1) An analogue of Trevisan’s result that relates the bipartite vertex expansion [Formula: see text] of a graph and the maximum reweighted lower spectral gap [Formula: see text] of the adjacency matrix. This implies the first approximation algorithm for bipartite vertex expansion. (2) An analogue of higher-order Cheeger’s inequalities that relates the [Formula: see text]-way vertex expansion [Formula: see text] of a graph and the maximum reweighted [Formula: see text]th smallest eigenvalue [Formula: see text] of the Laplacian matrix. This implies the first approximation algorithm for [Formula: see text]-way vertex expansion. (3) An analogue of improved Cheeger’s inequality that relates the vertex expansion [Formula: see text] and the reweighted eigenvalues [Formula: see text] and [Formula: see text]. This provides an improved bound for [Formula: see text] using [Formula: see text], when the [Formula: see text]-way vertex expansion [Formula: see text] is large for a small [Formula: see text]. Finally, inspired by this connection, we present negative evidence to the [Formula: see text]-polytope edge expansion conjecture by Mihail and Vazirani. We construct [Formula: see text]-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these [Formula: see text]-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of [Formula: see text]-polytopes.
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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 | 0.004 | 0.020 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.003 | 0.008 |
| Insufficient payload (model declined to judge) | 0.018 | 0.003 |
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".