Planar Novikov-Shubin invariant for adjacency matrices of structured directed dense random graphs
Torben Krüger, David Renfrew
TL;DR
The paper analyzes the accumulation of eigenvalues near the origin for adjacency matrices of dense directed stochastic block models by employing Girko's Hermitization and operator-valued free probability to derive a Matrix Dyson Equation for the Hermitized operator. It proves that the Brown measure density has a finite planar Novikov–Shubin invariant $c_{ m NS}$ when the connection-probability matrix $P$ has full support, and that $c_{ m NS}$ depends only on the zero-pattern of $P$ via a graph-based cycle metric with $c_{ m NS}=2oldsymbol{ abla}$, where in the two-block case $oldsymbol{ abla}= rac{1+ ext{ℓ}_1}{1+ ext{ℓ}_1+ ext{ℓ}_2}$. The singularity of the density at the origin is a power law $oldsymbol{ ho}(z) hicksim |z|^{-2(1-oldsymbol{ abla})}$, and the exponent is computed by solving a non-Hermitian min–max averaging equation derived from the Dyson equation in the $ au o0$ limit. The results yield a finite, computable invariant and provide a finite-step algorithm to determine it from the zero structure of $P$, with implications for invertibility and spectral concentration in large non-Hermitian random graphs.
Abstract
The Novikov-Shubin invariant associated to a graph provides information about the accumulation of eigenvalues of the corresponding adjacency matrix close to the origin. For a directed graph these eigenvalues lie in the complex plane and having a finite value for the planar Novikov-Shubin invariant indicates a polynomial behaviour of the eigenvalue density as a function of the distance to zero. We provide a complete description of these invariants for dense random digraphs with constant batch sizes, i.e. for the directed stochastic block model. The invariants depend only on which batches in the graph are connected by non-zero edge densities. We present an explicit finite step algorithm for their computation. For the proof we identify the asymptotic spectral density with the distribution of a $\mathbb{C}^K$-valued circular element in operator-valued free probability theory. We determine the spectral density in the bulk regime by solving the associated Dyson equation and infer the singular behaviour of this density close to the origin by determining the exponents associated to the power law with which the resolvent entries of the adjacency matrix that corresponds to the individual batches diverge to infinity or converge to zero.