# Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph boldsymbolβ-Model

arXiv: https://arxiv.org/abs/2307.02818 · HTML: https://mlsanity.com/paper/2307.02818/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 64 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 23 of 23)

- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Higher-order organization of complex networks](https://mlsanity.com/paper/1612.08447/) (2016)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration](https://mlsanity.com/paper/1909.06503/) (2019)
- [Consistency of Spectral Hypergraph Partitioning under Planted Partition Model](https://mlsanity.com/paper/1505.01582/) (2015)
- [Coauthorship and Citation Networks for Statisticians](https://mlsanity.com/paper/1410.2840/) (2014)
- [Hypergraph reconstruction from network data](https://mlsanity.com/paper/2008.04948/) (2020)
- [Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques](https://mlsanity.com/paper/1602.06516/) (2016)
- [Exact Recovery in the General Hypergraph Stochastic Block Model](https://mlsanity.com/paper/2105.04770/) (2021)
- [Community Recovery in Hypergraphs](https://mlsanity.com/paper/1709.03670/) (2017)
- [On the Minimax Misclassification Ratio of Hypergraph Community Detection](https://mlsanity.com/paper/1802.00926/) (2018)
- [Community detection in the sparse hypergraph stochastic block model](https://mlsanity.com/paper/1904.05981/) (2019)
- [Higher-order motif analysis in hypergraphs](https://mlsanity.com/paper/2108.03192/) (2021)
- [Maximum entropy distributions on graphs](https://mlsanity.com/paper/1301.3321/) (2013)
- [β models for random hypergraphs with a given degree sequence](https://mlsanity.com/paper/1407.1004/) (2014)
- [A Sparse Beta Regression Model for Network Analysis](https://mlsanity.com/paper/2010.13604/) (2020)
- [Nonparametric Modeling of Higher-Order Interactions via Hypergraphons](https://mlsanity.com/paper/2105.08678/) (2021)
- [Multiway Spherical Clustering via Degree-Corrected Tensor Block Models](https://mlsanity.com/paper/2201.07401/) (2022)
- [Analysis of Networks via the Sparse β-Model](https://mlsanity.com/paper/1908.03152/) (2019)
- [Hyperlink communities in higher-order networks](https://mlsanity.com/paper/2303.01385/) (2023)
- [Consistently estimating network statistics using Aggregated Relational Data](https://mlsanity.com/paper/1908.09881/) (2019)
- [L-2 Regularized maximum likelihood for β-model in large and sparse networks](https://mlsanity.com/paper/2110.11856/) (2021)

## Built on by (most cited first, 2 of 2)

- [Community Detection for Contextual-LSBM: Theoretical Limitations of Misclassification Rate and Efficient Algorithms](https://mlsanity.com/paper/2501.11139/) (2025)
- [Privacy utility trade offs for parameter estimation in degree heterogeneous higher order networks](https://mlsanity.com/paper/2602.03948/) (2026)

## Abstract (arXiv)

The boldsymbolβ-model for random graphs is commonly used for representing pairwise interactions in a network with degree heterogeneity. Going beyond pairwise interactions, Stasi et al. (2014) introduced the hypergraph boldsymbolβ-model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph boldsymbolβ-model with multiple layers, which allows for hyperedges of different sizes across the layers. To begin with, we derive the rates of convergence of the maximum likelihood (ML) estimate and establish their minimax rate optimality. We also derive the limiting distribution of the ML estimate and construct asymptotically valid confidence intervals for the model parameters. Next, we consider the goodness-of-fit problem in the hypergraph boldsymbolβ-model. Specifically, we establish the asymptotic normality of the likelihood ratio (LR) test under the null hypothesis, derive its detection threshold, and also its limiting power at the threshold. Interestingly, the detection threshold of the LR test turns out to be minimax optimal, that is, all tests are asymptotically powerless below this threshold. The theoretical results are further validated in numerical experiments. In addition to developing the theoretical framework for estimation and inference for hypergraph boldsymbolβ-models, the above results fill a number of gaps in the graph boldsymbolβ-model literature, such as the minimax optimality of the ML estimates and the non-null properties of the LR test, which, to the best of our knowledge, have not been studied before.