Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models
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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 {ϕ_u }_u=1^n with second moment Φ^(2). The intra-cluster connection probability for vertices u and v is ϕ_u ϕ_v/na and the inter-cluster connection probability is ϕ_u ϕ_v/nb. We show that with high probability, the following holds: The leading eigenvalue of the non-backtracking matrix B is asymptotic to ρ = a+b/2Φ^(2). The second eigenvalue is asymptotic to μ_2 = a-b/2Φ^(2) when μ_2^2 > ρ, but asymptotically bounded by √(ρ) when μ_2^2 ≤ρ. All the remaining eigenvalues are asymptotically bounded by √(ρ). 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 μ_2^2 > ρ. In a previous work we obtained that detection is impossible when μ_2^2 < ρ, 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 Erdős-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.
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