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Phase transition for the generalized two-community stochastic block model

Part of: Special matrices Theory of computing Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  31 July 2023

Sunmin Lee Open the ORCID record for Sunmin Lee [Opens in a new window]  and
Ji Oon Lee
Sunmin Lee*
Affiliation:
Korea Advanced Institute of Science and Technology (KAIST)
Ji Oon Lee*
Affiliation:
Korea Advanced Institute of Science and Technology (KAIST)
*
*Postal address: Department of Mathematical Sciences, KAIST, Daejeon 34141, South Korea.
*Postal address: Department of Mathematical Sciences, KAIST, Daejeon 34141, South Korea.
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Abstract

We study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.

Keywords

BBP-type transitionempirical spectral distributionWigner-type matrixhidden community modelunbalanced stochastic block model

MSC classification

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) 15B51: Stochastic matrices
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 385 - 400
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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