Community Detection with Contextual Multilayer Networks

Abstract

In this paper, we study community detection when we observe m sparse networks and a high dimensional covariate matrix, all encoding the same community structure among n subjects. In the asymptotic regime where the number of features p and the number of subjects n grows proportionally, we derive an exact formula of asymptotic minimum mean square error (MMSE) for estimating the common community structure in the balanced two block case. The formula implies the necessity of integrating information from multiple data sources. Consequently, it induces a sharp threshold of phase transition between the regime where detection (i.e., weak recovery) is possible and the regime where no procedure performs better than a random guess. The asymptotic MMSE depends on the covariate signal-to-noise ratio in a more subtle way than the phase transition threshold does. In the special case of m " 1, our asymptotic MMSE formula complements the pioneering work [12] which found the sharp threshold when m " 1.

A Non-Asymptotic Framework for Approximate Message Passing in Spiked Models

“…Originally introduced in the context of compressed sensing as a family of low-complexity iterative algorithms (Donoho et al, 2009), AMP lends it well to a wide spectrum of high-dimensional statistical problems, both as a class of efficient estimation algorithms and as a powerful theoretical machinery. Examples of this kind abound, including robust M-estimators Montanari, 2016, 2015), sparse linear regression (Bayati and Montanari, 2011b;Donoho et al, 2013;Bu et al, 2020;Li and Wei, 2021), generalized linear models Venkataramanan et al, 2021;Barbier et al, 2019), phase retrieval (Ma et al, 2018;Schniter and Rangan, 2014;Aubin et al, 2020), community detection (Deshpande et al, 2017;Ma and Nandy, 2021), structured matrix estimation and principal component analysis (PCA) (Rangan and Fletcher, 2012;Montanari and Venkataramanan, 2021;Deshpande and Montanari, 2014a;Mondelli and Venkataramanan, 2021), mean-field spin glass models (Sellke, 2021;Fan et al, 2022b;Fan and Wu, 2021), to name just a few. The interested reader is referred to Feng et al (2022) for a recent overview of AMP and its wide applicability.…”