Spectral analysis in the sparse regime

Determine whether local weak convergence techniques can be employed for kernel-based random graphs with \(\alpha>d\) to determine the properties of the limiting spectral measure, despite the underlying random graph not being tree-like.

Background

The paper restricts its main results to the dense regime $\alpha<d$. It identifies the regime $\alpha>d$ as substantially different because the graph is sparse, and notes that local weak convergence is a standard tool for studying spectra of sparse random graphs. However, the authors point out that the kernel-based random graph in this setting is not tree-like from the outset, creating uncertainty about whether those techniques apply. The limiting spectral measure and its properties therefore remain unresolved in this regime, which the authors defer to future work.