License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.44
URN: urn:nbn:de:0030-drops-81795
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8179/
Gulikers, Lennart ;
Lelarge, Marc ;
Massoulié, Laurent
Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models
Abstract
Motivated by community detection, we characterise the spectrum of the non-backtracking matrix B in the Degree-Corrected Stochastic Block Model.
Specifically, we consider a random graph on n vertices partitioned into two asymptotically equal-sized clusters. The vertices have i.i.d. weights {\phi_u}_{u=1}^n with second moment \PHItwo. The intra-cluster connection probability for vertices u and v is \frac{\phi_u \phi_v}{n}a and the inter-cluster connection probability is \frac{\phi_u \phi_v}{n}b.
We show that with high probability, the following holds: The leading eigenvalue of the non-backtracking matrix B is asymptotic to \rho = \frac{a+b}{2} \PHItwo. The second eigenvalue is asymptotic to \mu_2 = \frac{a-b}{2} \PHItwo when \mu_2^2 > \rho, but asymptotically bounded by \sqrt{\rho} when \mu_2^2 \leq \rho. All the remaining eigenvalues are asymptotically bounded by \sqrt{\rho}. As a result, a clustering positively-correlated with the true communities can be obtained based on the second eigenvector of B in the regime where \mu_2^2 > \rho.
In a previous work we obtained that detection is impossible when $\mu_2^2 \leq \rho,$ meaning that there occurs a phase-transition in the sparse regime of the Degree-Corrected Stochastic Block Model.
As a corollary, we obtain that Degree-Corrected Erdös-Rényi graphs asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan property.
A by-product of our proof is a weak law of large numbers for local-functionals on Degree-Corrected Stochastic Block Models, which could be of independent interest.
BibTeX - Entry
@InProceedings{gulikers_et_al:LIPIcs:2017:8179,
author = {Lennart Gulikers and Marc Lelarge and Laurent Massouli{\'e}},
title = {{Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models}},
booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages = {44:1--44:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-029-3},
ISSN = {1868-8969},
year = {2017},
volume = {67},
editor = {Christos H. Papadimitriou},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8179},
URN = {urn:nbn:de:0030-drops-81795},
doi = {10.4230/LIPIcs.ITCS.2017.44},
annote = {Keywords: Degree-Corrected Stochastic Block Model, Non-backtracking Matrix, Machine Learning, Social Networks}
}
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Keywords: |
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Degree-Corrected Stochastic Block Model, Non-backtracking Matrix, Machine Learning, Social Networks |
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Collection: |
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8th Innovations in Theoretical Computer Science Conference (ITCS 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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28.11.2017 |
