Community Detection With Contextual Multilayer Networks

In this paper, we study community detection when we observe <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> sparse networks and a high dimensional covariate matrix, all encoding the same community structure among <inline-formula> <...

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Vydáno v:IEEE transactions on information theory Ročník 69; číslo 5; s. 3203 - 3239
Hlavní autoři: Ma, Zongming, Nandy, Sagnik
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.05.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
approximate message passing
Asymptotic methods
Asymptotic properties
Clustering
contextual SBM
Estimation
integrative data analysis
Mathematical models
Message passing
multilayer network
Multilayers
Mutual information
Nonhomogeneous media
phase transition
Phase transitions
Signal to noise ratio
Social networking (online)
Soft sensors
stochastic block model
ISSN:0018-9448, 1557-9654
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Shrnutí:In this paper, we study community detection when we observe <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> sparse networks and a high dimensional covariate matrix, all encoding the same community structure among <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> subjects. In the asymptotic regime where the number of features <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> and the number of subjects <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> grow 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 using an orchestrated approximate message passing algorithm. 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 random guess. The asymptotic MMSE depends on the covariate signal-to-noise ratio in a more subtle way than the phase transition threshold. In the special case of <inline-formula> <tex-math notation="LaTeX">m=1 </tex-math></inline-formula>, our asymptotic MMSE formula complements the pioneering work Deshpande et al., (2018) which found the sharp threshold when <inline-formula> <tex-math notation="LaTeX">m=1 </tex-math></inline-formula>. A practical variant of the theoretically justified algorithm with spectral initialization leads to an estimator whose empirical MSEs closely approximate theoretical predictions over simulated examples.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3238352