# Inference for multiple heterogeneous networks with a common invariant subspace

arXiv: https://arxiv.org/abs/1906.10026 · HTML: https://mlsanity.com/paper/1906.10026/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2019.

## Built on by (most cited first, 30 of 39)

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- [Discovering Communication Pattern Shifts in Large-Scale Labeled Networks using Encoder Embedding and Vertex Dynamics](https://mlsanity.com/paper/2305.02381/) (2023)
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- [Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels](https://mlsanity.com/paper/2208.08638/) (2022)
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- [Privacy-Preserving Community Detection for Locally Distributed Multiple Networks](https://mlsanity.com/paper/2306.15709/) (2023)
- [Learning Joint and Individual Structure in Network Data with Covariates](https://mlsanity.com/paper/2406.08776/) (2024)
- [Principal Graph Encoder Embedding and Principal Community Detection](https://mlsanity.com/paper/2501.14939/) (2025)
- [Multivariate Analysis for Multiple Network Data via Semi-Symmetric Tensor PCA](https://mlsanity.com/paper/2202.04719/) (2022)
- [rMultiNet: An R Package For Multilayer Networks Analysis](https://mlsanity.com/paper/2302.04437/) (2023)
- [Nested stochastic block model for simultaneously clustering networks and nodes](https://mlsanity.com/paper/2307.09210/) (2023)
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- [Multiresolution Analysis and Statistical Thresholding on Dynamic Networks](https://mlsanity.com/paper/2506.01208/) (2025)
- [Community Detection in Weighted Multilayer Networks with Ambient Noise](https://mlsanity.com/paper/2103.00486/) (2021)

## Abstract (arXiv)

The development of models for multiple heterogeneous network data is of critical importance both in statistical network theory and across multiple application domains. Although single-graph inference is well-studied, multiple graph inference is largely unexplored, in part because of the challenges inherent in appropriately modeling graph differences and yet retaining sufficient model simplicity to render estimation feasible. This paper addresses exactly this gap, by introducing a new model, the common subspace independent-edge (COSIE) multiple random graph model, which describes a heterogeneous collection of networks with a shared latent structure on the vertices but potentially different connectivity patterns for each graph. The COSIE model encompasses many popular network representations, including the stochastic blockmodel. The model is both flexible enough to meaningfully account for important graph differences and tractable enough to allow for accurate inference in multiple networks. In particular, a joint spectral embedding of adjacency matrices - the multiple adjacency spectral embedding (MASE) - leads, in a COSIE model, to simultaneous consistent estimation of underlying parameters for each graph. Under mild additional assumptions, MASE estimates satisfy asymptotic normality and yield improvements for graph eigenvalue estimation and hypothesis testing. In both simulated and real data, the COSIE model and the MASE embedding can be deployed for a number of subsequent network inference tasks, including dimensionality reduction, classification, hypothesis testing and community detection. Specifically, when MASE is applied to a dataset of connectomes constructed through diffusion magnetic resonance imaging, the result is an accurate classification of brain scans by patient and a meaningful determination of heterogeneity across scans of different subjects.