Inference for multiple heterogeneous networks with a common invariant subspace

Abstract

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.

References in corpus (16)

• Community structure in social and biological networks
• Community Structure in Time-Dependent, Multiscale, and Multiplex Networks
• Stochastic blockmodels and community structure in networks
• Multirelational Organization of Large-scale Social Networks in an Online World
• Inferring the mesoscale structure of layered, edge-valued and time-varying networks
• Consistent estimation of dynamic and multi-layer block models
• On a 'Two Truths' Phenomenon in Spectral Graph Clustering
• Joint Embedding of Graphs
• Detecting Overlapping Communities in Networks Using Spectral Methods
• Symmetric Bilinear Regression for Signal Subgraph Estimation
• Spectral Clustering for Multiple Sparse Networks: I
• GraSPy: Graph Statistics in Python
• The Multiple Random Dot Product Graph Model
• Efficient Distributed Estimation of Inverse Covariance Matrices
• The eigenvalues of stochastic blockmodel graphs
• On the existence of flat orthogonal matrices

Cited by in corpus (15)

• Joint Inference of Multiple Graphs from Matrix Polynomials
• GraSPy: Graph Statistics in Python
• The multilayer random dot product graph
• Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities
• Clustering of heterogeneous populations of networks
• Link prediction in dynamic networks using random dot product graphs
• Compressing network populations with modal networks reveals structural diversity
• The Phantom Alignment Strength Conjecture: Practical use of graph matching alignment strength to indicate a meaningful graph match
• Bias-Variance Tradeoffs in Joint Spectral Embeddings
• Multiple Network Embedding for Anomaly Detection in Time Series of Graphs
• Simultaneous prediction and community detection for networks with application to neuroimaging
• Multiscale Comparative Connectomics
• Mutual Information in Community Detection with Covariate Information and Correlated Networks
• Nonparametric regression for multiple heterogeneous networks
• Valid Two-Sample Graph Testing via Optimal Transport Procrustes and Multiscale Graph Correlation with Applications in Connectomics