# Learning Joint and Individual Structure in Network Data with Covariates

arXiv: https://arxiv.org/abs/2406.08776 · HTML: https://mlsanity.com/paper/2406.08776/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 55 references that are in the map.
- First posted on arXiv 13 Jun 2024.

## Builds on (most cited first, 26 of 26)

- [Contextual Stochastic Block Models](https://mlsanity.com/paper/1807.09596/) (2018)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Layer aggregation and reducibility of multilayer interconnected networks](https://mlsanity.com/paper/1405.0425/) (2014)
- [Structure and inference in annotated networks](https://mlsanity.com/paper/1507.04001/) (2015)
- [Angle-Based Joint and Individual Variation Explained](https://mlsanity.com/paper/1704.02060/) (2017)
- [Covariate-assisted spectral clustering](https://mlsanity.com/paper/1411.2158/) (2014)
- [Spectral and matrix factorization methods for consistent community detection in multi-layer networks](https://mlsanity.com/paper/1704.07353/) (2017)
- [Community detection in node-attributed social networks: a survey](https://mlsanity.com/paper/1912.09816/) (2019)
- [Structural Learning and Integrative Decomposition of Multi-View Data](https://mlsanity.com/paper/1707.06573/) (2017)
- [sJIVE: Supervised Joint and Individual Variation Explained](https://mlsanity.com/paper/2102.13278/) (2021)
- [Latent space models for multiplex networks with shared structure](https://mlsanity.com/paper/2012.14409/) (2020)
- [Prediction models for network-linked data](https://mlsanity.com/paper/1602.01192/) (2016)
- [Covariate Regularized Community Detection in Sparse Graphs](https://mlsanity.com/paper/1607.02675/) (2016)
- [Joint Embedding of Graphs](https://mlsanity.com/paper/1703.03862/) (2017)
- [Community Detection in Networks with Node Features](https://mlsanity.com/paper/1509.01173/) (2015)
- [Pairwise Covariates-adjusted Block Model for Community Detection](https://mlsanity.com/paper/1807.03469/) (2018)
- [Linear regression and its inference on noisy network-linked data](https://mlsanity.com/paper/2007.00803/) (2020)
- [Data Integration Via Analysis of Subspaces (DIVAS)](https://mlsanity.com/paper/2212.00703/) (2022)
- [Testing and Modeling Dependencies Between a Network and Nodal Attributes](https://mlsanity.com/paper/1306.4708/) (2013)
- [Network Dependence Testing via Diffusion Maps and Distance-Based Correlations](https://mlsanity.com/paper/1703.10136/) (2017)
- [Double-matched matrix decomposition for multi-view data](https://mlsanity.com/paper/2105.03396/) (2021)
- [High-dimensional Factor Analysis for Network-linked Data](https://mlsanity.com/paper/2403.17121/) (2024)

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

- [Multi-task Learning for Heterogeneous Multi-source Block-Wise Missing Data](https://mlsanity.com/paper/2505.24413/) (2025)
- [Covariate-assisted graph matching](https://mlsanity.com/paper/2512.11761/) (2025)

## Abstract (arXiv)

Datasets consisting of a network and covariates associated with its vertices have become ubiquitous. One problem pertaining to this type of data is to identify information unique to the network, information unique to the vertex covariates and information that is shared between the network and the vertex covariates. Existing techniques for network data and vertex covariates focus on capturing structure that is shared but are usually not able to differentiate structure that is unique to each dataset. This work formulates a low-rank model that simultaneously captures joint and individual information in network data with vertex covariates. A two-step estimation procedure is proposed, composed of an efficient spectral method followed by a refinement optimization step. Theoretically, we show that the spectral method is able to consistently recover the joint and individual components under a general signal-plus-noise model. Simulations and real data examples demonstrate the ability of the methods to recover accurate and interpretable components. In particular, the application of the methodology to a food trade network between countries with economic, developmental and geographical country-level indicators as covariates yields joint and individual factors that explain the trading patterns.