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arxiv: 2606.06482 · v2 · pith:K7APUBSYnew · submitted 2026-06-04 · 📊 stat.ME · math.ST· stat.TH

Two-Sample Hypothesis Testing for Subspace Equality in Network Data

Rajdeep Brahma , Joshua Agterberg , Yuguo Chen This is my paper

Pith reviewed 2026-06-28 00:08 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords network hypothesis testingsubspace equalitystochastic blockmodeltwo-sample testeigenvector perturbationasymptotic normalitygraph testingprojection matrix
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The pith

A test for subspace equality between two networks has its centered and scaled statistic converge in distribution to Gaussian when average expected degree grows at least logarithmically in the number of vertices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a hypothesis test for whether two networks share the same underlying subspace, covering shared community structure in stochastic blockmodels or mixed-membership variants even when edge probabilities differ. The test statistic is the Frobenius norm of the difference between the leading subspace projection matrices from each network's adjacency matrix. The central result establishes that after suitable centering and scaling this statistic converges in distribution to a Gaussian random variable provided the average expected degree grows at least logarithmically with the number of vertices. Estimators for the asymptotic mean and variance are shown to be consistent under stronger signal conditions, and the local power of the test is derived for sufficiently dense networks. The proof relies on a supporting limit theorem for the difference between empirical and true eigenvector projections, which may have uses beyond the two-sample setting.

Core claim

The paper claims that the Frobenius norm of the difference of the leading subspace projection matrices from two networks, after appropriate centering and scaling, converges in distribution to a Gaussian random variable whenever the average expected degree grows at least logarithmically in the number of vertices. This convergence holds for general subspace equality testing and in particular for stochastic blockmodels and mixed-membership stochastic blockmodels with identical communities but possibly different edge probabilities. The result is obtained via a one-sample limit theorem on the projection difference between empirical and population eigenvectors, which is presented as potentially of

What carries the argument

The test statistic given by the Frobenius norm of the difference between the two leading subspace projection matrices, supported by a limit theorem for the difference between empirical and true eigenvector projections.

If this is right

  • The test applies directly to stochastic blockmodels and mixed-membership stochastic blockmodels that share communities but have different edge probabilities.
  • Estimators for the asymptotic mean and variance remain consistent when the signal strength satisfies a stricter condition than the basic logarithmic-degree requirement.
  • The test possesses nontrivial local power against local alternatives when the networks are sufficiently dense.
  • The one-sample eigenvector-projection limit theorem can be applied to single-network inference problems that involve eigenvector differences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The logarithmic-degree threshold suggests the procedure may remain valid in moderately sparse regimes that lie between the dense and extremely sparse settings studied in earlier network-testing literature.
  • The same projection-difference machinery could be adapted to test subspace equality in degree-corrected blockmodels or other latent-space network models without altering the core convergence argument.
  • Applied researchers could compare networks observed at different times or under different experimental conditions to detect whether community structure has changed while allowing for shifts in overall edge density.

Load-bearing premise

The average expected degree grows at least logarithmically in the number of vertices.

What would settle it

Simulations in which the centered and scaled test statistic deviates from a standard normal distribution when the average expected degree grows exactly logarithmically with the number of vertices would falsify the claimed convergence.

read the original abstract

In many settings one is often interested in determining whether two networks share some joint structural connectivity patterns such as communities. However, while communities may be shared across networks, edge probabilities may differ significantly. Therefore, in this paper we consider testing a general null hypothesis that two networks have the same underlying subspace, which in particular includes the setting that communities are the same for either stochastic blockmodels or mixed-membership stochastic blockmodels (even if edge probabilities are different). We propose a test statistic based on the Frobenius norm of the difference of the leading subspace projection matrices, and we prove that our test statistic, after appropriate centering and scaling, converges in distribution to a Gaussian random variable as long as the average expected degree grows at least logarithmically in the number of vertices. We then provide estimators for the asymptotic mean and variance and show consistency under a stronger signal condition, and we give the local power of our test when the networks are sufficiently dense. Our theoretical results are based on a limit theorem for the projection difference of empirical and true eigenvectors which can also be viewed as the one-sample version of our test statistic, and this result may be of independent interest. We demonstrate our results through numerical simulations and an application to US Flight data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a two-sample test for equality of leading subspaces between two networks (e.g., stochastic block models or mixed-membership variants sharing communities but with possibly different edge probabilities). The test statistic is the Frobenius norm of the difference between the empirical leading subspace projection matrices. The central theoretical contribution is a limit theorem establishing that, after centering and scaling, this statistic converges in distribution to a Gaussian random variable provided the average expected degree satisfies d_n = Ω(log n). The authors supply consistent estimators for the asymptotic mean and variance under a stronger signal regime, derive the local power of the resulting test, and note that the one-sample projection-difference limit theorem may be of independent interest. Results are illustrated via simulations and a US Flight data example.

Significance. If the stated limit theorem and consistency results hold, the work supplies a practical, asymptotically justified procedure for testing structural similarity across networks without requiring identical edge probabilities. The explicit degree-growth condition and the one-sample projection limit theorem constitute clear technical strengths; the approach is non-parametric with respect to the subspace and directly addresses a common applied setting in network analysis.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'stronger signal condition' for variance estimation is left unspecified; the manuscript should state the precise regime (e.g., a concrete lower bound on d_n relative to n) already in the abstract so that readers can immediately assess applicability.
  2. [Introduction] The manuscript should include a brief comparison, in the introduction or discussion, of the log n degree condition with existing results on eigenvector perturbation or subspace recovery in the network literature (e.g., results requiring d_n = Ω(log n / log log n) or polynomial growth).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the accurate summary of the main contributions, and the recommendation for minor revision. We are pleased that the referee highlights the practical utility of the non-parametric test, the explicit degree-growth condition, and the independent interest of the one-sample projection limit theorem.

Circularity Check

0 steps flagged

No circularity; central limit theorem is a direct proof

full rationale

The paper states and proves a new limit theorem for the projection difference of empirical and true eigenvectors (one-sample version of the test statistic) under the explicit condition that average expected degree grows at least logarithmically. No equations reduce the claimed convergence or variance estimators to a fitted quantity by construction, no self-citations are invoked as load-bearing uniqueness results, and no ansatz is smuggled or known result renamed. The derivation chain consists of standard concentration arguments for network eigenvectors followed by explicit centering/scaling, all presented as self-contained results of independent interest. This is the normal case of a theoretical statistics paper whose main claim is an asymptotic statement proved from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The convergence claim rests on standard asymptotic theory for random matrices and eigenvectors in sparse graphs plus the domain assumption that the networks follow stochastic block or mixed-membership stochastic block models with a shared subspace. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard central limit theorems and concentration results for eigenvectors of random adjacency matrices apply under the stated degree growth condition.
    Invoked to obtain the Gaussian limit after centering and scaling.
  • domain assumption The two networks are generated from SBM or MMSBM with identical leading subspace but possibly different edge probabilities.
    This is the modeling framework under which the null hypothesis of subspace equality is defined and tested.

pith-pipeline@v0.9.1-grok · 5752 in / 1363 out tokens · 27412 ms · 2026-06-28T00:08:40.595904+00:00 · methodology

discussion (0)

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