# Hypothesis Testing for Equality of Latent Positions in Random Graphs

arXiv: https://arxiv.org/abs/2105.10838 · HTML: https://mlsanity.com/paper/2105.10838/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 67 references that are in the map.
- First posted on arXiv 23 May 2021.

## Builds on (most cited first, 19 of 19)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Network cross-validation by edge sampling](https://mlsanity.com/paper/1612.04717/) (2016)
- [Perfect Clustering for Stochastic Blockmodel Graphs via Adjacency Spectral Embedding](https://mlsanity.com/paper/1310.0532/) (2013)
- [Network Cross-Validation for Determining the Number of Communities in Network Data](https://mlsanity.com/paper/1411.1715/) (2014)
- [Hypothesis Testing For Network Data in Functional Neuroimaging](https://mlsanity.com/paper/1407.5525/) (2014)
- [A goodness-of-fit test for stochastic block models](https://mlsanity.com/paper/1412.4857/) (2014)
- [Robust and computationally feasible community detection in the presence of arbitrary outlier nodes](https://mlsanity.com/paper/1404.6000/) (2014)
- [Limit theorems for eigenvectors of the normalized Laplacian for random graphs](https://mlsanity.com/paper/1607.08601/) (2016)
- [Hypothesis Testing for Automated Community Detection in Networks](https://mlsanity.com/paper/1311.2694/) (2013)
- [Network Enhancement: a general method to denoise weighted biological networks](https://mlsanity.com/paper/1805.03327/) (2018)
- [Two-sample Hypothesis Testing for Inhomogeneous Random Graphs](https://mlsanity.com/paper/1707.00833/) (2017)
- [Network Representation Using Graph Root Distributions](https://mlsanity.com/paper/1802.09684/) (2018)
- [Determining the Number of Communities in Degree-corrected Stochastic Block Models](https://mlsanity.com/paper/1809.01028/) (2018)
- [Vertex nomination schemes for membership prediction](https://mlsanity.com/paper/1312.2638/) (2013)
- [Efficient Estimation for Random Dot Product Graphs via a One-step Procedure](https://mlsanity.com/paper/1910.04333/) (2019)
- [Vertex Nomination, Consistent Estimation, and Adversarial Modification](https://mlsanity.com/paper/1905.01776/) (2019)
- [SIMPLE: Statistical Inference on Membership Profiles in Large Networks](https://mlsanity.com/paper/1910.01734/) (2019)

## Built on by (most cited first, 5 of 5)

- [Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels](https://mlsanity.com/paper/2208.08638/) (2022)
- [Detection of Model-based Planted Pseudo-cliques in Random Dot Product Graphs by the Adjacency Spectral Embedding and the Graph Encoder Embedding](https://mlsanity.com/paper/2312.11054/) (2023)
- [Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Data](https://mlsanity.com/paper/2412.19796/) (2024)
- [Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)](https://mlsanity.com/paper/2506.02825/) (2025)
- [Matching and mixing: Matchability of graphs under Markovian error](https://mlsanity.com/paper/2601.20020/) (2026)

## Abstract (arXiv)

We consider the hypothesis testing problem that two vertices i and j of a generalized random dot product graph have the same latent positions, possibly up to scaling. Special cases of this hypothesis test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors, or testing whether two vertices in a popularity adjusted block model have the same community assignment. We propose several test statistics based on the empirical Mahalanobis distances between the ith and jth rows of either the adjacency or the normalized Laplacian spectral embedding of the graph. We show that, under mild conditions, these test statistics have limiting chi-square distributions under both the null and local alternative hypothesis, and we derived explicit expressions for the non-centrality parameters under the local alternative. Using these limit results, we address the model selection problems including choosing between the standard stochastic block model and its degree-corrected variant, and choosing between the ER model and stochastic block model. The effectiveness of our proposed tests are illustrated via both simulation studies and real data applications.