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Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models

Lennart Gulikers 1, 2, 3 Marc Lelarge 2 Laurent Massoulié 1, 3
2 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
3 INFINE - INFormation NEtworks
Inria Saclay - Ile de France
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 equal-sized clusters. The vertices have i.i.d. weights $\{ \phi_u \}_{u=1}^n$ with second moment $\Phi^{(2)}$. 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 $\rho = \frac{a+b}{2} \Phi^{(2)}$. The second eigenvalue is asymptotic to $\mu_2 = \frac{a-b}{2} \Phi^{(2)}$ 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 $\mu_2^2 < \rho,$ 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 Erd\H{o}s-R\'enyi 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.
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Contributor : Lennart Gulikers <>
Submitted on : Tuesday, October 24, 2017 - 3:59:32 PM
Last modification on : Tuesday, May 4, 2021 - 2:06:02 PM

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  • HAL Id : hal-01622719, version 1
  • ARXIV : 1609.02487



Lennart Gulikers, Marc Lelarge, Laurent Massoulié. Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models. ITCS 2017 - 8th Innovations in Theoretical Computer Science, Jan 2017, Berkeley, United States. pp.1-52. ⟨hal-01622719⟩



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