Section: New Results

Non-backtracking spectrum of degree-corrected stochastic block models

[25] 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 ${\left\{{\phi }_u\right\}}_{u=1}^n$ with second moment ${\Phi }^{\left(2\right)}$. 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 $\rho =\frac{a+b}{2}{\Phi }^{\left(2\right)}$. The second eigenvalue is asymptotic to ${\mu }_2=\frac{a-b}{2}{\Phi }^{\left(2\right)}$ when ${\mu }_2^2>\rho$, but asymptotically bounded by $\sqrt{\rho }$ when ${\mu }_2^2\le \rho$. 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 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.