  Spectral clustering is one of the most popular, yet still incompletely
understood, methods for community detection on graphs. This article studies
spectral clustering based on the Bethe-Hessian matrix $H_r = (r^2-1)I_n + D-rA$
for sparse heterogeneous graphs (following the degree-corrected stochastic
block model) in a two-class setting. For a specific value $r = \zeta$,
clustering is shown to be insensitive to the degree heterogeneity. We then
study the behavior of the informative eigenvector of $H_{\zeta}$ and, as a
result, predict the clustering accuracy. The article concludes with an overview
of the generalization to more than two classes along with extensive simulations
on synthetic and real networks corroborating our findings.
