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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

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.

Lennart Gulikers, Marc Lelarge, and Laurent Massoulié. Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 44:1-44:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gulikers_et_al:LIPIcs.ITCS.2017.44, author = {Gulikers, Lennart and Lelarge, Marc and Massouli\'{e}, Laurent}, 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 = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.44}, 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} }